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\title[Pretorsion theories and image-closed subcategories]
{Pretorsion theories and image-closed subcategories}
\author[H.\ Enomoto]{Haruhisa Enomoto}
\address{Parakeet Inc., Japan}
\email{haruhisa.enomoto.math@gmail.com}
\subjclass[2020]{16G10, 16G20, 05A15}
\keywords{pretorsion theory, image-closed subcategory, convex geometry,
interval module, Genocchi number, Gandhi polynomial}
\date{\today}

\begin{document}

\begin{abstract}
A pretorsion theory generalizes a torsion pair by allowing its two
components to have a nonzero intersection, with morphisms factoring through
the intersection playing the role of zero morphisms.  In the module
category of a finite-dimensional algebra, the first and second components
of pretorsion theories are known to be the quotient-closed and
submodule-closed subcategories, respectively.  We show that the intersections of the two components
are precisely the image-closed subcategories, namely those
containing the image of every morphism between their objects, and that the
pretorsion theories form a complete lattice.  For a linearly oriented
quiver of type A, we prove that the numbers of pretorsion theories and of
image-closed subcategories are the Genocchi and median Genocchi numbers,
respectively.  Counting pretorsion theories by the number of indecomposable
modules in their intersection, and image-closed subcategories by their
number of indecomposable modules, yields the Gandhi polynomial refinements
of Han and Zeng.
\end{abstract}

\maketitle
\begingroup
\footnotesize
\setcounter{tocdepth}{2}
\tableofcontents
\endgroup

\section{Introduction}\label{sec:introduction}

Let \(\Lambda\) be a finite-dimensional algebra over a field \(\kk\), and
consider full, additive subcategories of \(\mod\Lambda\) closed under
isomorphisms and direct summands.  We ask which subcategories can occur as
the intersection \(\TT\cap\FF\) of the two components of a pretorsion
theory \((\TT,\FF)\) in \(\mod\Lambda\).  Recall that a torsion pair
\((\TT,\FF)\) in \(\mod\Lambda\) satisfies \(\Hom_\Lambda(\TT,\FF)=0\), and
every module \(X\in\mod\Lambda\) fits into a short exact sequence
\(0\to T\to X\to F\to0\) with \(T\in\TT\) and \(F\in\FF\), where \(T\to X\)
is a kernel of \(X\to F\) and \(X\to F\) is its cokernel; in particular,
\(\TT\cap\FF=0\).  Pretorsion theories relax this orthogonality condition
by allowing the intersection \(\ZZZ=\TT\cap\FF\) to be nonzero: morphisms
factoring through \(\ZZZ\) replace zero morphisms both in the orthogonality
condition between \(\TT\) and \(\FF\) and in the kernel and cokernel
universal properties (Definition~\ref{def:pretorsion}).  Facchini and
Finocchiaro \cite{FF} introduced this notion for categories without a zero
object—starting from the category of nonempty preordered sets, where
equivalence relations and partial orders form the two components and sets
equipped with equality form their intersection—and Facchini, Finocchiaro,
and Gran \cite[Section~2]{FFG} developed the theory for general categories.

Characterizing the intersections \(\TT\cap\FF\) in \(\mod\Lambda\) brings
together the study of quotient-closed and submodule-closed subcategories in
the author's earlier work \cite{EnomotoEquidistribution} and that of
image-closed subcategories in joint work with Sakai \cite{EnomotoSakai}.
Indeed, Campanini, Fedele, and Y{\i}ld{\i}r{\i}m
\cite[Lemma~2.6, Proposition~2.8, and Section~3.2]{CFY} proved that the
subcategories occurring as first and second components of pretorsion
theories in \(\mod\Lambda\) are precisely the quotient-closed and
submodule-closed subcategories, respectively.  Earlier, Campanini and
Fedele \cite[Theorem~4.2]{CampaniniFedele} constructed pretorsion theories
whose intersections are prescribed Serre subcategories.  Here we determine
all possible intersections, allowing subcategories that are not Serre.

A subcategory \(\ZZZ\subseteq\mod\Lambda\) is \emph{image-closed} if
\(\Image f\in\ZZZ\) for every morphism \(f\colon X\to Y\) with
\(X,Y\in\ZZZ\).  Write \(\Gen\ZZZ\) and \(\Cogen\ZZZ\) for the subcategories
consisting of the quotients and the submodules, respectively, of finite
direct sums of objects of \(\ZZZ\).  Auslander and Smal{\o}
\cite[Proposition~4.10(i)]{AuslanderSmalo} proved that a subcategory
\(\ZZZ\) is image-closed if and only if \(\ZZZ=\Gen\ZZZ\cap\Cogen\ZZZ\).
Our first main result shows that although \((\Gen\ZZZ,\Cogen\ZZZ)\) need
not itself be a pretorsion theory (Example~\ref{ex:semisimple-middle}), every image-closed subcategory \(\ZZZ\)
is the intersection \(\TT\cap\FF\) of some pretorsion theory \((\TT,\FF)\).

\begin{theoremi}[{= Theorem~\ref{thm:image-middle}}]
\label{thm:intro-main}
Let \(\Lambda\) be a finite-dimensional algebra, and let \(\ZZZ\) be a subcategory
of \(\mod \Lambda\).  The following are equivalent.
\begin{enumerate}
\item \(\ZZZ\) is image-closed.
\item \(\ZZZ=\Gen\ZZZ\cap\Cogen\ZZZ\).
\item \(\ZZZ=\TT\cap\FF\) for a pretorsion theory \((\TT,\FF)\) in \(\mod \Lambda\).
\end{enumerate}
\end{theoremi}

Theorem~\ref{thm:intro-main} may be compared with a result of the author and
Sakai \cite[Proposition~2.3]{EnomotoSakai}, who proved that the subcategories
of \(\mod\Lambda\) closed under both images and extensions are precisely the
intersections of a torsion class and a torsion-free class (which need not
form a torsion pair).

To construct a pretorsion theory with a prescribed image-closed
intersection, we use Theorem~\ref{thm:trace-reject}, which characterizes
when a quotient-closed subcategory \(\TT\) and a submodule-closed
subcategory \(\FF\) form a pretorsion theory by comparing, for each module,
its largest submodule in \(\TT\) (its trace) with its largest quotient in
\(\FF\) (the quotient by its reject).

We next study the poset \(\PT(\Lambda)\) of pretorsion theories in
\(\mod\Lambda\), ordered by inclusion of first components and reverse
inclusion of second components.  Complementing the complete lattices of
individual components studied in \cite[Sections~3.1--3.2]{CFY}, we prove
that \(\PT(\Lambda)\) is a complete lattice and describe its meet and join
(Theorem~\ref{thm:PT-lattice}).  A pretorsion theory with intersection
\(\ZZZ\) is called a \emph{realization} of \(\ZZZ\).  For each image-closed
subcategory \(\ZZZ\), its realizations form a complete lattice with
greatest and least elements (Corollaries~\ref{cor:greatest-realization}
and~\ref{cor:least-realization}), whose nonempty meets and joins agree with
those in \(\PT(\Lambda)\)
(Proposition~\ref{prop:fixed-intersection-lattice}).  In particular, for
\(\ZZZ=0\), the realizations are precisely the torsion pairs, which form a
complete sublattice containing the least and greatest elements of
\(\PT(\Lambda)\) and isomorphic to the lattice of torsion classes
(Corollary~\ref{cor:torsion-sublattice}).

To study the poset \(\LI(\Lambda)\) of image-closed subcategories of
\(\mod\Lambda\), ordered by inclusion, we use convex geometry
\cite{EdelmanJamison} to describe its generating sets and covering
relations.  Write \(\ind\Lambda\) for a set of representatives of the
isomorphism classes of indecomposable \(\Lambda\)-modules, and \(\ind\CC\)
for those belonging to \(\CC\).  The \emph{size} of \(\CC\) is
\(\lvert\ind\CC\rvert\), the number of isomorphism classes of its
indecomposable objects.  Sending
a subset of \(\ind\Lambda\) to the indecomposable objects of the smallest
image-closed subcategory containing it defines a closure operator \(\icl\) on
\(\ind\Lambda\).  Extending the corresponding result for quotient-closed and
submodule-closed subcategories in \cite{EnomotoEquidistribution}, we show
that \((\ind\Lambda,\icl)\) is a finitary \emph{convex geometry}
(Theorem~\ref{thm:icl-convex}): we have \(\icl(\varnothing)=\varnothing\),
the closure of any subset is the union of the
closures of its finite subsets, and \(\icl\) satisfies the anti-exchange
property that if \(\CC\) is image-closed and
\(X,Y\in\ind\Lambda\setminus\ind\CC\) are distinct, then
\(X\in\icl(\ind\CC\cup\{Y\})\) implies \(Y\notin\icl(\ind\CC\cup\{X\})\).

As in the geometry of convex hulls and their extreme points, this
convex-geometric structure has direct consequences for \(\LI(\Lambda)\)
(Corollary~\ref{cor:LI-structure}).  Every image-closed subcategory \(\ZZZ\)
of finite size has a unique smallest generating set of indecomposables,
consisting of its \emph{extreme points}—the indecomposables in \(\ZZZ\)
that do not belong to the image closure of the remaining ones.  Proposition~\ref{prop:extreme-points}
identifies this set as the union of the splitting projective and splitting
injective objects of Auslander and Smal{\o} \cite[Section~2]{AuslanderSmalo}:
an indecomposable \(M\in\ZZZ\) is splitting projective if every epimorphism
to \(M\) from an object of \(\ZZZ\) splits, and splitting injective if every
monomorphism from \(M\) to an object of \(\ZZZ\) splits.  The
existence of this smallest generating set also follows from
\cite[Proposition~2.1 and Corollary~2.4(b),(d)]{AuslanderSmalo}, together
with the description of image closure as \(\Gen\cap\Cogen\).  Moreover,
every covering relation in \(\LI(\Lambda)\) adds a single indecomposable
module, so \(\LI(\Lambda)\) is graded by size whenever \(\Lambda\) is
representation-finite.

\subsection*{The linearly oriented quiver of type \texorpdfstring{\(A\)}{A}}

Campanini, Fedele, and Y{\i}ld{\i}r{\i}m \cite[Appendix~A.2]{CFY}
listed the \(17\) pretorsion theories for the linearly oriented quiver
of type \(A_2\).  In the second half of this paper, we consider the path
algebra
\[
 \Lambda_n=\kk(1\longrightarrow2\longrightarrow\cdots\longrightarrow n)
\]
of a linearly oriented quiver of type \(A_n\) (with \(\Lambda_0=0\)), and
determine the numbers of pretorsion theories and of image-closed
subcategories in \(\mod\Lambda_n\) for every \(n\geq 0\), together with their
natural \(q\)-refinements.  These enumeration formulas are established only
for the linearly oriented quiver: unlike the number of torsion pairs, the numbers
of pretorsion theories and of image-closed subcategories for a quiver of
type \(A_n\) depend on the orientation
(Remark~\ref{rem:orientation-counts}).  For small values of \(n\), the
cardinalities of \(\PT(\Lambda_n)\) and \(\LI(\Lambda_n)\) are given in the
following table.
\begin{center}
\begin{tabular}{c|rrrrrr}
\toprule
\(n\)&\(0\)&\(1\)&\(2\)&\(3\)&\(4\)&\(5\)\\
\midrule
\(\lvert\PT(\Lambda_n)\rvert\)&\(1\)&\(3\)&\(17\)&\(155\)&\(2073\)&\(38227\)\\
\(\lvert\LI(\Lambda_n)\rvert\)&\(1\)&\(2\)&\(8\)&\(56\)&\(608\)&\(9440\)\\
\bottomrule
\end{tabular}
\end{center}
These are the sequences A110501 and A005439 in the OEIS
\cite{OEISGenocchi,OEISMedian}, given by the positive Genocchi numbers
\(g_{2n+4}\) and the median Genocchi numbers \(H_{2n+3}\).  Following Han and Zeng
\cite[p.~63]{HanZengMedian}, the positive Genocchi numbers
\(g_{2i}\) and the median Genocchi numbers \(H_{2m+1}\) are defined by
\begin{equation}\label{eq:genocchi-egf}
 \begin{gathered}
 \frac{2t}{e^t+1}=\sum_{i\geq0}G_i\frac{t^i}{i!},\qquad g_{2i}=|G_{2i}|,\\
 H_{2m+1}=\sum_{0\leq i\leq (m-1)/2}(-1)^i g_{2m-2i}\binom{m}{2i+1}
 \quad(m\geq1).
 \end{gathered}
\end{equation}
To refine these numbers to polynomials in \(q\), we recall the \(q\)-Gandhi
polynomials introduced by Han and Zeng \cite{HanZeng,HanZengMedian}.

\begin{definition}[{\cite[p.~120, Equations~(3)--(4)]{HanZeng}}]
\label{def:gandhi-polynomials}
For a polynomial \(f\in\ZZ[q,x]\), put
\begin{equation}\label{eq:intro-delta}
                  \Delta_q f(x)=
                  \frac{f(1+qx)-f(x)}{1+(q-1)x}.
\end{equation}
The \emph{\(q\)-Gandhi polynomials of the first and of the second kind} are
defined by \(B_1(x,q)=C_1(x,q)=1\) and, for \(m\geq2\),
\begin{align}
 B_m(x,q)&=\Delta_q\bigl(x^2B_{m-1}(x,q)\bigr),\label{eq:intro-B}\\
 C_m(x,q)&=(1+qx)\Delta_q\bigl(xC_{m-1}(x,q)\bigr).\label{eq:intro-C}
\end{align}
\end{definition}

Since \(1+(q-1)x=(1+qx)-x\), the operator \(\Delta_q\) preserves
\(\ZZ[q,x]\) and specializes at \(q=1\) to the forward difference operator
\(f(x)\mapsto f(x+1)-f(x)\).  The classical specializations are
\(B_m(1,1)=g_{2m+2}\) \cite[p.~120, Equation~(1)]{HanZeng} and
\(C_m(1,1)=H_{2m+1}\) \cite[pp.~63--64, Equation~(2)]{HanZengMedian}.

The quotient--submodule equidistribution conjecture of
\cite{EnomotoEquidistribution} compares the numbers of quotient-closed and
submodule-closed subcategories of each size for a representation-finite
algebra, predicting that they agree.  Counting subcategories by size in
this way motivates our \(q\)-refinement, which enumerates pretorsion
theories by the size of their intersections and image-closed subcategories
by their size.

\begin{theoremi}[{= Theorems~\ref{thm:pretorsion-gandhi} and
\ref{thm:image-gandhi}}]
\label{thm:intro-enumeration}
For every \(n\geq0\),
\begin{align*}
 \sum_{(\TT,\FF)\in\PT(\Lambda_n)}
 q^{\lvert\ind(\TT\cap\FF)\rvert}&=B_{n+1}(1,q),\\
 \sum_{\CC\in\LI(\Lambda_n)}q^{\lvert\ind\CC\rvert}&=C_{n+1}(1,q).
\end{align*}
\end{theoremi}

Since \(\LI(\Lambda_n)\) is graded by size, \(C_{n+1}(1,q)\) is its
rank-generating polynomial.  At \(q=1\), Theorem~\ref{thm:intro-enumeration}
gives \(\lvert\PT(\Lambda_n)\rvert=g_{2n+4}\) and
\(\lvert\LI(\Lambda_n)\rvert=H_{2n+3}\).

To prove Theorem~\ref{thm:intro-enumeration}, we describe both families of
subcategories combinatorially.  Each indecomposable \(\Lambda_n\)-module is
determined by its support interval in \(\{1,\ldots,n\}\), and we show in
Section~\ref{sec:intervals} that the image-closed subcategories of
\(\mod\Lambda_n\) correspond bijectively to the families of intervals in
\(\{1,\ldots,n\}\) closed under nonempty intersections
(Theorem~\ref{thm:intersection-criterion}).  Pilaud
\cite[p.~3]{PilaudPlumbing} also obtained the count \(H_{2n+3}\) for these
intersection-closed interval families.  In Section~\ref{sec:pretorsion-type-a},
we parametrize quotient-closed and submodule-closed subcategories of
\(\mod\Lambda_n\) by endpoints recording how far the support intervals
extend when one endpoint is fixed, and characterize which pairs form a pretorsion theory
(Theorem~\ref{thm:staircase-pair}).  Adjoining one vertex at a time in
Sections~\ref{sec:first-gandhi} and~\ref{sec:image-closed}, we then
establish two-variable refinements of Theorem~\ref{thm:intro-enumeration}
(Theorems~\ref{thm:pretorsion-gandhi} and~\ref{thm:image-gandhi}) in which
the auxiliary variable \(x\) records the number of indecomposable modules
whose support ends at vertex \(n\), in \(\TT\) for \(B_{n+1}(x,q)\) and in
\(\CC\) for \(C_{n+1}(x,q)\).

For comparison, a \emph{Genocchi permutation} of \(\{1,\ldots,2m\}\) is a
permutation \(\sigma\) satisfying \(\sigma(2i-1)>2i-1\) and
\(\sigma(2i)\leq2i\) for \(1\leq i\leq m\).  Writing
\(\sigma_{\mathrm{odd}}\) and \(\sigma_{\mathrm{even}}\) for its subwords
in odd and even positions and \(\operatorname{inv}\) for the number of
inversions, Han and Zeng \cite[Proposition~19 and Corollary~21]{HanZeng}
show that \(B_m(1,q)\) is the sum, over all Genocchi permutations \(\sigma\)
of \(\{1,\ldots,2m\}\), of the weights
\[
 q^{m^2-\sum_{i=1}^m\sigma(2i)
       -\operatorname{inv}(\sigma_{\mathrm{odd}})
       -\operatorname{inv}(\sigma_{\mathrm{even}})},
\]
and that restricting the sum to fixed-point-free Genocchi permutations gives
\(C_m(1,q)\).  Our identifications in
Theorem~\ref{thm:intro-enumeration} are proved by recurrences, and no
statistic-preserving bijection between our subcategories and these
permutation models is known to us.

\begin{org}
Section~\ref{sec:pretorsion-general} recalls pretorsion theories and proves
the trace--reject criterion (Theorem~\ref{thm:trace-reject}), and
Section~\ref{sec:lattice} shows that \(\PT(\Lambda)\) is a complete lattice
(Theorem~\ref{thm:PT-lattice}).  Section~\ref{sec:middles} proves
Theorem~\ref{thm:intro-main} and studies the convex geometry of image
closure.  Section~\ref{sec:intervals} describes morphisms between
indecomposable \(\Lambda_n\)-modules and characterizes the image-closed
subcategories of \(\mod\Lambda_n\) by their support intervals.
Section~\ref{sec:pretorsion-type-a} classifies the pretorsion theories in
\(\mod\Lambda_n\).
Sections~\ref{sec:first-gandhi} and~\ref{sec:image-closed} prove the two
formulas of Theorem~\ref{thm:intro-enumeration}.
\end{org}

\begin{conv}
Throughout this paper, \(\kk\) is an arbitrary field and \(\Lambda\) is a
finite-dimensional \(\kk\)-algebra.  All modules are finite-dimensional
right modules, and \(\mod\Lambda\) denotes the category of
finite-dimensional right \(\Lambda\)-modules.  Paths in a quiver are
composed from left to right, so \(\Lambda_n\)-modules are representations of
the quiver \(1\to 2\to\cdots\to n\).

All subcategories of \(\mod\Lambda\) are assumed to be full, additive, and
closed under isomorphisms and direct summands; \(0\) denotes the zero
subcategory.  For a collection of modules \(\mathcal M\), let
\(\add\mathcal M\) denote the subcategory of direct summands of finite
direct sums of modules in \(\mathcal M\).  We write \(\ind\Lambda\) for a
fixed set of representatives of the isomorphism classes of indecomposable
\(\Lambda\)-modules, and \(\ind\CC\) for the subset of representatives
belonging to a subcategory \(\CC\subseteq\mod\Lambda\).

For integers \(a\) and \(d\), the interval \([a,d]\) denotes the set
\(\{a,a+1,\ldots,d\}\), which is empty when \(a>d\).  Inequalities between
integer vectors are understood coordinatewise.
\end{conv}

\begin{aiuse}
The research was done under the author's direction using GPT-5.6-Sol,
which found and developed most of the results and proofs presented here;
GPT-6-Astra contributed further results and checked the mathematics.
The manuscript was drafted and revised with GPT-6-Astra, Claude Opus~5.5,
and Gemini~4 Argon, which were used heavily in writing, revising, and
polishing the text. The author directed the research and the writing and
is responsible for the mathematical content and the final manuscript.
\end{aiuse}
\section{Pretorsion theories, trace, and reject}
\label{sec:pretorsion-general}

In this section, we recall the definition of a pretorsion theory in
\(\mod\Lambda\) and characterize it in terms of the trace and the reject of
its two components (Theorem~\ref{thm:trace-reject}).

\begin{definition}\label{def:image-closed}
A subcategory \(\CC\subseteq\mod \Lambda\) is \emph{quotient-closed} if
every quotient of an object of \(\CC\) belongs to \(\CC\), and it is
\emph{submodule-closed} if every submodule of an object of \(\CC\) belongs to
\(\CC\).  It is \emph{image-closed} if \(\Image f\in\CC\) for every morphism
\(f\colon X\to Y\) with \(X,Y\in\CC\).  Write \(\LQ(\Lambda)\), \(\LS(\Lambda)\), and
\(\LI(\Lambda)\) for the sets of quotient-closed, submodule-closed, and image-closed
subcategories of \(\mod \Lambda\), each ordered by inclusion.
\end{definition}

Every quotient-closed subcategory and every submodule-closed subcategory is
image-closed, because the image of a morphism \(f\colon X\to Y\) is both a
quotient of \(X\) and a submodule of \(Y\).  More generally, if
\(\TT\in\LQ(\Lambda)\) and \(\FF\in\LS(\Lambda)\), every morphism \(f\) from an
object of \(\TT\) to an object of \(\FF\) satisfies
\begin{equation}\label{eq:image-in-intersection}
                         \Image f\in\TT\cap\FF.
\end{equation}
In particular, \(\TT\cap\FF\in\LI(\Lambda)\).

We now recall the relative kernel and cokernel conditions that define a
pretorsion theory.

\begin{definition}[{\cite[Definition~2.6]{FFG}, specialized to \(\mod \Lambda\)}]
\label{def:pretorsion}
Fix a subcategory \(\ZZZ\subseteq\mod \Lambda\).  A morphism is
\emph{\(\ZZZ\)-trivial} if it factors through an object of \(\ZZZ\).  For
composable morphisms
\[
                         T\xrightarrow{u}X\xrightarrow{v}F,
\]
the morphism \(u\) is a \emph{\(\ZZZ\)-prekernel} of \(v\) if \(vu\) is
\(\ZZZ\)-trivial and every morphism \(u'\colon Y\to X\) for which \(vu'\) is
\(\ZZZ\)-trivial factors as \(u'=uh\) for a unique morphism
\(h\colon Y\to T\).  Dually, \(v\) is a \emph{\(\ZZZ\)-precokernel} of \(u\)
if \(vu\) is \(\ZZZ\)-trivial and every morphism \(v'\colon X\to Y\) for
which \(v'u\) is \(\ZZZ\)-trivial factors as \(v'=hv\) for a unique morphism
\(h\colon F\to Y\).  In diagrams,
\[
\begin{tikzcd}[row sep=large,column sep=large]
 T\arrow[r,"u"] & X\arrow[r,"v"] & F\\
 & Y\arrow[ul,dashed,"h"]\arrow[u,swap,"u'"] &
\end{tikzcd}
\qquad\qquad
\begin{tikzcd}[row sep=large,column sep=large]
 T\arrow[r,"u"] & X\arrow[r,"v"]\arrow[d,swap,"v'"] & F\arrow[dl,dashed,"h"]\\
 & Y &
\end{tikzcd}
\]
where \(vu\) is \(\ZZZ\)-trivial, as are \(vu'\) on the left and \(v'u\) on
the right, and \(h\) is the unique morphism making the respective triangle
commute.  The sequence \(T\xrightarrow{u}X\xrightarrow{v}F\) is \emph{short
\(\ZZZ\)-preexact} if \(u\) is a \(\ZZZ\)-prekernel of \(v\) and \(v\) is a
\(\ZZZ\)-precokernel of \(u\).

A \emph{pretorsion theory} in \(\mod \Lambda\) is a pair \((\TT,\FF)\) of
subcategories of \(\mod \Lambda\) such that, for \(\ZZZ=\TT\cap\FF\), the following
two conditions hold.
\begin{enumerate}
\item Every morphism from an object of \(\TT\) to an object of \(\FF\) is
      \(\ZZZ\)-trivial.
\item Every module \(X\in\mod\Lambda\) occurs in a short \(\ZZZ\)-preexact
      sequence \(T\to X\to F\) with \(T\in\TT\) and \(F\in\FF\).
\end{enumerate}
\end{definition}

The definition of a pretorsion theory in \cite[Definition~2.6]{FFG} and
\cite[Section~2]{CFY} requires only full subcategories closed under
isomorphisms, without assuming closure under finite direct sums or direct
summands.  The following result of Campanini, Fedele, and Y{\i}ld{\i}r{\i}m
\cite{CFY} shows that these closure properties hold automatically in
\(\mod\Lambda\).

\begin{proposition}[{\cite[Lemma~2.6 and Proposition~2.8; dual statement in Section~3.2]{CFY}}]
\label{prop:pretorsion-closure}
Let \((\TT,\FF)\) be a pretorsion theory of full subcategories closed under
isomorphisms in \(\mod\Lambda\).  Then \(\TT\) is closed under quotients and finite
direct sums, and \(\FF\) is closed under submodules and finite direct sums.
\end{proposition}

Since any direct summand of a module is both a quotient and a submodule of
it, the components \(\TT\) and \(\FF\) are also closed under direct summands,
and hence satisfy our standing subcategory conventions.  Thus the two
conventions give the same pretorsion theories in \(\mod\Lambda\).

We write \(\PT(\Lambda)\) for the set of pretorsion theories in
\(\mod\Lambda\), ordered by
\begin{equation}\label{eq:PT-order}
 (\TT,\FF)\leq(\TT',\FF')
 \quad\Longleftrightarrow\quad
 \TT\subseteq\TT'\text{ and }\FF\supseteq\FF'.
\end{equation}
When \(\TT\cap\FF=0\), Definition~\ref{def:pretorsion} reduces to the
classical notion of a torsion pair.

\begin{definition}\label{def:torsion-pair}
A \emph{torsion pair} in \(\mod \Lambda\) is a pair \((\TT,\FF)\) of
subcategories such that \(\Hom_{\Lambda}(\TT,\FF)=0\) and every module \(X\) occurs in
a short exact sequence
\[
 0\longrightarrow T\longrightarrow X\longrightarrow F\longrightarrow0
\]
with \(T\in\TT\) and \(F\in\FF\).
\end{definition}

The following observation identifies the torsion pairs among pretorsion
theories.

\begin{proposition}\label{prop:zero-middle-torsion}
A pair \((\TT,\FF)\) of subcategories of \(\mod \Lambda\) is a torsion pair if and
only if it is a pretorsion theory with \(\TT\cap\FF=0\).
\end{proposition}

\begin{proof}
\((\Rightarrow)\): If \((\TT,\FF)\) is a torsion pair, then
\(\Hom_{\Lambda}(\TT,\FF)=0\) implies \(\TT\cap\FF=0\), and the defining
short exact sequences are short \(0\)-preexact sequences.

\((\Leftarrow)\): When \(\ZZZ=0\), a morphism is \(\ZZZ\)-trivial if and
only if it is zero, so \(\ZZZ\)-prekernels and \(\ZZZ\)-precokernels
coincide with ordinary kernels and cokernels.  Thus a pretorsion theory with
\(\TT\cap\FF=0\) is a torsion pair.
\end{proof}

Over \(\Lambda_1=\kk\), one can list the pretorsion theories directly.

\begin{example}\label{ex:A1-pretorsion}
The category \(\mod\Lambda_1\) has a single indecomposable module up to
isomorphism, namely the simple module \(S\), so each of \(\TT\) and \(\FF\)
is either \(0\) or \(\mod\Lambda_1\).  The pair \((0,0)\) has intersection
\(0\), so by Proposition~\ref{prop:zero-middle-torsion} it would have to be
a torsion pair; however, \(S\) does not fit into a short exact sequence
\(0\to 0\to S\to 0\to 0\).  For \((\mod\Lambda_1,\mod\Lambda_1)\), we have
\(\ZZZ=\mod\Lambda_1\), so every morphism is \(\ZZZ\)-trivial and the
identity maps give a short \(\ZZZ\)-preexact sequence
\[
 X\xrightarrow{1_X}X\xrightarrow{1_X}X
\]
for every \(X\in\mod\Lambda_1\).  The remaining two pairs
\((0,\mod\Lambda_1)\) and \((\mod\Lambda_1,0)\) are the ordinary torsion
pairs, which are pretorsion theories by
Proposition~\ref{prop:zero-middle-torsion}.  Thus \(\PT(\Lambda_1)\) consists
of the three pairs
\[
 (0,\mod \Lambda_1),\qquad(\mod \Lambda_1,\mod \Lambda_1),\qquad(\mod \Lambda_1,0),
\]
whose intersections are \(0\), \(\mod \Lambda_1\), and \(0\), respectively.
\end{example}

We next recall the trace and reject constructions of Auslander and Smal{\o}
\cite[Proposition~4.8 and Corollary~4.9]{AuslanderSmalo}.

\begin{definition}\label{def:trace-reject}
Let \(\CC\subseteq\mod \Lambda\) be a subcategory and let
\(X\in\mod \Lambda\).  The \emph{trace of \(\CC\) in \(X\)} and the \emph{reject of
\(\CC\) in \(X\)} are the submodules
\[
 t_{\CC}(X)=\sum_{Y\in\CC,\ f\colon Y\to X}\Image f,
 \qquad
 r_{\CC}(X)=\bigcap_{Y\in\CC,\ g\colon X\to Y}\Ker g
\]
of \(X\), where the sum and the intersection range over all objects
\(Y\in\CC\) and all morphisms \(f\colon Y\to X\) and \(g\colon X\to Y\).
\end{definition}

\begin{remark}\label{rem:duality}
The standard duality \(\DDual=\Hom_{\kk}(-,\kk)\) sends
\(\ZZZ\)-prekernels to \(\DDual\ZZZ\)-precokernels and conversely, and
therefore induces an order-reversing bijection
\[
 \PT(\Lambda)\longrightarrow\PT(\Lambda^{\op}),\qquad
 (\TT,\FF)\longmapsto(\DDual\FF,\DDual\TT),
\]
with respect to~\eqref{eq:PT-order}.  We will use this bijection to deduce
dual assertions in the proofs below.  The functor \(\DDual\) also exchanges
trace submodules with reject quotients:
\[
 \DDual(t_{\CC}(X))\cong\DDual X/r_{\DDual\CC}(\DDual X),\qquad
 \DDual(X/r_{\CC}(X))\cong t_{\DDual\CC}(\DDual X).
\]
These identifications follow by dualizing the sums of images and
intersections of kernels in Definition~\ref{def:trace-reject}.
\end{remark}

Although Definition~\ref{def:trace-reject} takes the sum and the
intersection over all morphisms, the following observation for modules of
finite length shows that finitely many images or kernels suffice.

\begin{lemma}\label{lem:finite-intersection}
Every sum or intersection of submodules of a module of finite length is the
sum or intersection of finitely many of them.
\end{lemma}

\begin{proof}
This follows from the ascending and descending chain conditions on submodules.
\end{proof}

We collect the basic properties of the trace and the reject when \(\TT\) is
quotient-closed and \(\FF\) is submodule-closed.

\begin{proposition}[{\cite[Proposition~4.8 and Corollary~4.9]{AuslanderSmalo}}]
\label{prop:trace-reject-properties}
Let \(\TT\in\LQ(\Lambda)\), let \(\FF\in\LS(\Lambda)\), and let
\(X\in\mod \Lambda\).
\begin{enumerate}
\item \(t_{\TT}(X)\) is the largest submodule of \(X\) belonging to \(\TT\).
\item \(X/r_{\FF}(X)\) is the largest quotient of \(X\) belonging to \(\FF\).
\item \(X\in\TT\) if and only if \(t_{\TT}(X)=X\), and \(X\in\FF\) if and
      only if \(r_{\FF}(X)=0\).
\item Both operations commute with finite direct sums:
      \begin{equation}\label{eq:trace-reject-sums}
       t_{\TT}(X_1\oplus X_2)=t_{\TT}(X_1)\oplus t_{\TT}(X_2),
       \qquad
       r_{\FF}(X_1\oplus X_2)=r_{\FF}(X_1)\oplus r_{\FF}(X_2).
      \end{equation}
\item \(t_{\TT}(W)\subseteq t_{\TT}(X)\) and
      \(r_{\FF}(W)\subseteq r_{\FF}(X)\) whenever \(W\subseteq X\).
\end{enumerate}
\end{proposition}

\begin{proof}
(1) This is \cite[Proposition~4.8(a)]{AuslanderSmalo}: by
Lemma~\ref{lem:finite-intersection}, the trace is a sum of finitely many
images of morphisms from objects of \(\TT\), hence a quotient of a finite
direct sum of objects of \(\TT\), so it belongs to \(\TT\) and contains
every \(\TT\)-submodule of \(X\).

(2) This is dual to (1), or \cite[Proposition~4.8(b)]{AuslanderSmalo}.

(3) This follows immediately from (1) and (2).

(4) By \cite[Corollary~4.9]{AuslanderSmalo}, \(t_{\TT}\) is right adjoint
to the inclusion \(\TT\hookrightarrow\mod\Lambda\), and
\(X\mapsto X/r_{\FF}(X)\) is left adjoint to
\(\FF\hookrightarrow\mod\Lambda\).  These additive functors preserve finite
biproducts; taking kernels of the quotient maps gives the assertion for
\(r_{\FF}\).

(5) Composing any morphism \(Y\to W\) (\(Y\in\TT\)) with the inclusion
\(W\hookrightarrow X\) shows that \(t_{\TT}(W)\subseteq t_{\TT}(X)\).  For
the reject, restricting any morphism \(g\colon X\to Y\) (\(Y\in\FF\)) to
\(W\) gives a morphism \(g|_W\colon W\to Y\), so
\(r_{\FF}(W)\subseteq\Ker(g|_W)\subseteq\Ker g\); intersecting over all such
\(g\) yields \(r_{\FF}(W)\subseteq r_{\FF}(X)\).
\end{proof}

To describe short preexact sequences in terms of submodules and quotients,
we use the following property of their maps.

\begin{lemma}[{\cite[Propositions~2.1--2.2]{FFG}}]
\label{lem:relative-mono-epi}
Let \(\ZZZ\subseteq\mod \Lambda\) be a subcategory.  Every
\(\ZZZ\)-prekernel in \(\mod \Lambda\) is a monomorphism, and every
\(\ZZZ\)-precokernel is an epimorphism.
\end{lemma}

Not every pair \((\TT,\FF)\in\LQ(\Lambda)\times\LS(\Lambda)\) is a
pretorsion theory.  For any such pair, condition~(1) of
Definition~\ref{def:pretorsion} holds automatically: by~\eqref{eq:image-in-intersection},
each map factors through its image in \(\TT\cap\FF\).  Thus only the existence of
short \((\TT\cap\FF)\)-preexact sequences in condition~(2) is at issue.  The
following theorem characterizes the pairs that form pretorsion theories by
two equalities relating the trace and the reject.

\begin{theorem}\label{thm:trace-reject}
Let \(\TT\in\LQ(\Lambda)\) and \(\FF\in\LS(\Lambda)\), and write \(t=t_{\TT}\) and
\(r=r_{\FF}\).  Then \((\TT,\FF)\) is a pretorsion theory
if and only if
\begin{equation}\label{eq:trace-reject}
                  r(X)=r(t(X)),\qquad
                  t(X/r(X))=t(X)/r(X)
\end{equation}
for every \(X\in\mod \Lambda\).  Here the first equality implies
\(r(X)\subseteq t(X)\), so that \(t(X)/r(X)\) is defined and the second
equality is an equality of submodules of \(X/r(X)\).  When~\eqref{eq:trace-reject}
holds, the subquotient \(t(X)/r(X)\) belongs to \(\TT\cap\FF\), and the canonical
sequence
\[
                         t(X)\longrightarrow X\longrightarrow X/r(X)
\]
is short \((\TT\cap\FF)\)-preexact.
\end{theorem}

\begin{proof}
Put \(\ZZZ=\TT\cap\FF\).  Under the identifications in
Remark~\ref{rem:duality}, dualizing either equality in
\eqref{eq:trace-reject} gives the other equality for
\((\DDual\FF,\DDual\TT)\).  We use this to pass between the two universal
properties.

\((\Leftarrow)\): Suppose that~\eqref{eq:trace-reject} holds for \(X\in\mod\Lambda\).  The
first equality gives \(r(X)=r(t(X))\subseteq t(X)\).  Moreover,
\(t(X)/r(X)\) belongs to \(\TT\) as a quotient of \(t(X)\), and to
\(\FF\) as a submodule of \(X/r(X)\).  Hence \(t(X)/r(X)\in\ZZZ\).

Let \(i\colon t(X)\hookrightarrow X\) be the inclusion and
\(p\colon X\twoheadrightarrow X/r(X)\) the quotient map.  The composite
\(pi\) has image \(t(X)/r(X)\in\ZZZ\) and is therefore \(\ZZZ\)-trivial.  If
\(f\colon W\to X\) is a morphism such that \(pf\) is \(\ZZZ\)-trivial, choose
a factorization \(pf=b_fa_f\) through an object \(N_f\in\ZZZ\):
\[
\begin{tikzcd}[row sep=large,column sep=large]
 W\arrow[r,dashed]\arrow[d,swap,"a_f"]\arrow[rr,bend left=22,"f"]
 & t(X)\arrow[r,hook,"i"] & X\arrow[d,twoheadrightarrow,"p"]\\
 N_f\arrow[rr,swap,"b_f"] & & X/r(X)
\end{tikzcd}
\]
Since \(N_f\in\ZZZ\subseteq\TT\) and \(\TT\) is quotient-closed, we have
\(\Image b_f\in\TT\).  By Proposition~\ref{prop:trace-reject-properties}(1)
and the second equality in~\eqref{eq:trace-reject}, it follows that
\[
               \Image(pf)\subseteq\Image b_f\subseteq t(X/r(X))=t(X)/r(X).
\]
Consequently, \(\Image f\subseteq p^{-1}(t(X)/r(X))=t(X)\), so \(f\) factors
through \(i\), uniquely because \(i\) is a monomorphism.  Thus \(i\) is a
\(\ZZZ\)-prekernel of \(p\).

Dually, \(p\) is a \(\ZZZ\)-precokernel of \(i\), by
Remark~\ref{rem:duality} and the first equality in~\eqref{eq:trace-reject}.
Thus the canonical sequence is short \(\ZZZ\)-preexact.

\((\Rightarrow)\): Conversely, assume that \((\TT,\FF)\) is a pretorsion theory, and choose a
short \(\ZZZ\)-preexact sequence
\[
                         T_X\xrightarrow{u}X\xrightarrow{v}F_X
\]
with \(T_X\in\TT\) and \(F_X\in\FF\).  By Lemma~\ref{lem:relative-mono-epi},
\(u\) is a monomorphism and \(v\) is an epimorphism, so we may identify
\(T_X\) with \(\Image u\subseteq X\) and \(F_X\) with \(X/\Ker v\).

Since \(\Image u\) is a quotient of \(T_X\in\TT\), we have
\(\Image u\in\TT\), and hence \(\Image u\subseteq t(X)\).  On the other hand, the
restriction of \(v\) to \(t(X)\) is a morphism from \(t(X)\in\TT\) to
\(F_X\in\FF\), so it is \(\ZZZ\)-trivial by condition~(1) of
Definition~\ref{def:pretorsion}.  By the \(\ZZZ\)-prekernel property of
\(u\), the inclusion \(t(X)\hookrightarrow X\) factors through \(u\), giving
\(t(X)\subseteq\Image u\).  Thus \(\Image u=t(X)\), and \(u\) is isomorphic
over \(X\) to the inclusion \(i\colon t(X)\hookrightarrow X\).

Dually, \(\Ker v=r(X)\), so we may take
\(u=i\colon t(X)\hookrightarrow X\) and
\(v=p\colon X\twoheadrightarrow X/r(X)\).

Put \(U=t(X/r(X))\) and \(P=p^{-1}(U)\).  Since \(U\in\TT\) and
\(U\subseteq X/r(X)\in\FF\), we have \(U\in\ZZZ\).  The composite
\(P\hookrightarrow X\xrightarrow{p}X/r(X)\) has image \(U\in\ZZZ\) and is
therefore \(\ZZZ\)-trivial; since \(i\colon t(X)\hookrightarrow X\) is a
\(\ZZZ\)-prekernel of \(p\), the inclusion \(P\hookrightarrow X\) factors
through \(i\), giving \(P\subseteq t(X)\).  Conversely, \(p(t(X))\) is a
quotient of \(t(X)\in\TT\), so it belongs to \(\TT\) and is therefore
contained in \(U\); hence \(t(X)\subseteq p^{-1}(U)=P\).  Thus
\(P=t(X)\), so \(r(X)\subseteq t(X)\) and
\(t(X/r(X))=U=t(X)/r(X)\).  Dually, because \(p\) is a
\(\ZZZ\)-precokernel of \(i\), we obtain \(r(X)=r(t(X))\),
completing~\eqref{eq:trace-reject}.
\end{proof}

For pretorsion theories in a general category, Facchini, Finocchiaro, and
Gran \cite[Proposition~2.7]{FFG} showed that any two of the three classes
\(\TT,\FF,\ZZZ\) determine the third.  Specializing
\cite[Proposition~3.9]{FFG} and its dual to \(\mod\Lambda\) yields the
following reconstruction formulas in terms of the trace and the reject.
We will use them to construct a pair with a prescribed intersection in
Section~\ref{sec:middles}.

\begin{corollary}[{\cite[Proposition~3.9 and its dual]{FFG}}]
\label{cor:reconstruction}
If \((\TT,\FF)\) is a pretorsion theory and \(\ZZZ=\TT\cap\FF\), then
\begin{equation}\label{eq:reconstruction}
 \FF=\{Y\in\mod\Lambda:t_{\TT}(Y)\in\ZZZ\},
 \qquad
 \TT=\{X\in\mod\Lambda:X/r_{\FF}(X)\in\ZZZ\}.
\end{equation}
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:pretorsion-closure}, \(\TT\in\LQ(\Lambda)\) and
\(\FF\in\LS(\Lambda)\), so Proposition~\ref{prop:trace-reject-properties} and
Theorem~\ref{thm:trace-reject} apply to \((\TT,\FF)\).
For \(Y\in\FF\), we have \(r_{\FF}(Y)=0\), so
\(r_{\FF}(t_{\TT}(Y))=r_{\FF}(Y)=0\) by Theorem~\ref{thm:trace-reject}, and
hence \(t_{\TT}(Y)\in\TT\cap\FF=\ZZZ\).  Conversely, if
\(t_{\TT}(Y)\in\ZZZ\subseteq\FF\), then
\(r_{\FF}(Y)=r_{\FF}(t_{\TT}(Y))=0\), so \(Y\in\FF\).  This proves the first
equality in~\eqref{eq:reconstruction}.

The second equality follows dually.
\end{proof}
\section{The lattice of pretorsion theories}
\label{sec:lattice}

In this section, we use Theorem~\ref{thm:trace-reject} to prove that the
poset \(\PT(\Lambda)\), ordered by~\eqref{eq:PT-order}, is a complete
lattice whose nonempty meets and joins preserve a fixed intersection
(Theorem~\ref{thm:PT-lattice} and
Proposition~\ref{prop:fixed-intersection-lattice}).

Campanini, Fedele, and Y{\i}ld{\i}r{\i}m
\cite[Proposition~3.4 and Section~3.2]{CFY} showed that the individual first
and second components of pretorsion theories, namely \(\LQ(\Lambda)\) and
\(\LS(\Lambda)\), form complete lattices under inclusion.  Here we study the
pairs \((\TT,\FF)\) forming pretorsion theories.

By Theorem~\ref{thm:trace-reject}, \((\TT,\mod\Lambda)\) is a pretorsion
theory for every \(\TT\in\LQ(\Lambda)\): the reject \(r_{\mod\Lambda}(X)\)
vanishes for every module \(X\), so both equalities in~\eqref{eq:trace-reject}
hold trivially.  Dually, \((\mod\Lambda,\FF)\) is a pretorsion theory for
every \(\FF\in\LS(\Lambda)\).  Together with
Proposition~\ref{prop:pretorsion-closure}, these examples show that the
possible first and second components of pretorsion theories are precisely
the members of \(\LQ(\Lambda)\) and \(\LS(\Lambda)\), respectively.  In particular, \((0,\mod\Lambda)\) and \((\mod\Lambda,0)\) are the least and
greatest elements of \(\PT(\Lambda)\), respectively.

To construct the meet of a nonempty family \((\TT_i,\FF_i)_{i\in I}\) in
\(\PT(\Lambda)\), we pair the intersection \(\bigcap_{i\in I}\TT_i\) of its
first components with the smallest subcategory \(\FF\) containing every
\(\FF_i\) such that \((\bigcap_i\TT_i,\FF)\) is a pretorsion theory.  The
existence of such a smallest \(\FF\) follows once we show that, for a fixed
\(\TT\in\LQ(\Lambda)\), the subcategories \(\FF\) with
\((\TT,\FF)\in\PT(\Lambda)\) are closed under arbitrary intersections
(Proposition~\ref{prop:partner-intersection}), since \(\mod\Lambda\) always
belongs to this collection.  To prove that the resulting pair is the
greatest lower bound, we then show in Proposition~\ref{prop:lower-bound}
that if \((\TT'',\FF'')\) is any lower bound of the family, then
\((\bigcap_{i\in I}\TT_i,\FF'')\) is also a pretorsion theory.

For two second components \(\FF_1,\FF_2\) with the same first component,
the sum \(r_{\FF_1}(X)+r_{\FF_2}(X)\) is contained in
\(r_{\FF_1\cap\FF_2}(X)\).  We will repeatedly divide by these sums until
the quotient belongs to both second components.  The next lemma shows
that, for \(R(X)=r_{\FF_1}(X)+r_{\FF_2}(X)\), the trace on \(X/R(X)\)
is \(t_{\TT}(X)/R(X)\).  This allows us to recover the second equality
in~\eqref{eq:trace-reject} at \(X\) from the same equality at \(X/R(X)\).

\begin{lemma}\label{lem:trace-quotient}
Let \((\TT,\FF_1)\) and \((\TT,\FF_2)\) be pretorsion theories, write
\(t=t_{\TT}\) and \(r_i=r_{\FF_i}\), and put \(R(X)=r_1(X)+r_2(X)\) for a
module \(X\).  Then \(R(X)\subseteq t(X)\) and
\[
 t\bigl(X/R(X)\bigr)=t(X)/R(X).
\]
\end{lemma}

\begin{proof}
By Theorem~\ref{thm:trace-reject}, we have \(r_i(X)\subseteq t(X)\) for
\(i=1,2\), so \(R(X)\subseteq t(X)\).  Since \(t(X)/R(X)\) is a quotient of
\(t(X)\in\TT\), it belongs to \(\TT\) and is therefore contained in
\(t(X/R(X))\) by Proposition~\ref{prop:trace-reject-properties}(1).

For the reverse inclusion, we first divide by \(r_1(X)\) and compare
\(X/R(X)\) with a further quotient whose trace is known from
Theorem~\ref{thm:trace-reject}.  Put \(Y=X/r_1(X)\), let
\(p\colon X\twoheadrightarrow Y\) be the quotient map, and put
\(K=p(R(X))=p(r_2(X))\), so that \(X/R(X)\cong Y/K\).  Every morphism
\(g\colon Y\to G\) with \(G\in\FF_2\) satisfies \(gp(r_2(X))=0\), so
\(K\subseteq r_2(Y)\).  Applying Theorem~\ref{thm:trace-reject} to
\((\TT,\FF_1)\) at \(X\) and to \((\TT,\FF_2)\) at \(Y\) gives
\[
 t(Y)=t(X)/r_1(X),\qquad
 r_2(Y)\subseteq t(Y),\qquad
 t\bigl(Y/r_2(Y)\bigr)=t(Y)/r_2(Y).
\]

Let \(q\colon Y/K\twoheadrightarrow Y/r_2(Y)\) be the canonical quotient
map.  The image \(q(t(Y/K))\) belongs to \(\TT\) and hence is contained in
\(t(Y/r_2(Y))=t(Y)/r_2(Y)\).  Thus the restriction of \(q\) fits into the
commutative diagram
\[
\begin{tikzcd}[column sep=large]
 t(Y/K)\arrow[r]\arrow[d,hook]
   & t(Y)/r_2(Y)\arrow[d,hook]\\
 Y/K\arrow[r,twoheadrightarrow,"q"] & Y/r_2(Y).
\end{tikzcd}
\]
Taking the inverse image under \(q\) and using \(r_2(Y)\subseteq t(Y)\) yields
\[
 t(Y/K)\subseteq q^{-1}\bigl(t(Y)/r_2(Y)\bigr)=\bigl(t(Y)+r_2(Y)\bigr)/K=t(Y)/K.
\]
Under the canonical isomorphism \(Y/K\cong X/R(X)\), the submodule
\[
 t(Y)/K=\bigl(t(X)/r_1(X)\bigr)\big/\bigl(R(X)/r_1(X)\bigr)
\]
corresponds to \(t(X)/R(X)\).  Consequently,
\(t(X/R(X))\subseteq t(X)/R(X)\), which gives the reverse inclusion.
\end{proof}

We now show that the second components forming a pretorsion theory with a
fixed \(\TT\) are closed under arbitrary intersections.

\begin{proposition}\label{prop:partner-intersection}
Let \(\TT\) be quotient-closed and let \((\TT,\FF_i)\), \(i\in I\), be a
family of pretorsion theories.  Then
\((\TT,\bigcap_{i\in I}\FF_i)\) is a pretorsion theory.
\end{proposition}

\begin{proof}
For the empty family, \((\TT,\mod\Lambda)\) is a pretorsion theory by
Theorem~\ref{thm:trace-reject}, since \(r_{\mod\Lambda}=0\).
Assume \(I\neq\varnothing\).
We first prove closure under finite intersections by repeatedly dividing
out two rejects, and then pass to arbitrary intersections using finite length.

\medskip
\noindent\textbf{Step 1: Finite intersections.}
Let \(I=\{1,2\}\), and put \(\FF'=\FF_1\cap\FF_2\), \(r'=r_{\FF'}\),
\(t=t_{\TT}\), \(r_i=r_{\FF_i}\), and \(R(M)=r_1(M)+r_2(M)\) for
\(M\in\mod\Lambda\).  Every morphism from \(M\) to an object of \(\FF'\)
vanishes on both \(r_1(M)\) and \(r_2(M)\), and hence on \(R(M)\).
Composing with the quotient map \(\pi_M\colon M\twoheadrightarrow M/R(M)\)
therefore identifies the morphisms from \(M/R(M)\) to objects of \(\FF'\)
with those from \(M\), so that
\begin{equation}\label{eq:transport}
 r'(M)=\pi_M^{-1}\bigl(r'(M/R(M))\bigr),\qquad
 M/r'(M)\cong\bigl(M/R(M)\bigr)\big/r'\bigl(M/R(M)\bigr).
\end{equation}
We verify~\eqref{eq:trace-reject} for \((\TT,\FF')\) at every module \(X\)
by induction on \(\dim_{\kk} X\), using~\eqref{eq:transport} to reduce to
\(X/R(X)\) whenever \(R(X)\neq0\).  If \(R(X)=0\), then
\(r_1(X)=r_2(X)=0\), so \(X\in\FF_1\cap\FF_2=\FF'\) by
Proposition~\ref{prop:trace-reject-properties}(3); since \(\FF'\) is
submodule-closed, we also have \(t(X)\in\FF'\), so
\(r'(X)=r'(t(X))=0\) and both equalities in~\eqref{eq:trace-reject} hold.

Suppose now that \(R(X)\neq 0\), put \(V=X/R(X)\), and let
\(\pi\colon X\twoheadrightarrow V\) be the quotient map.  Since
\(\dim_{\kk} V<\dim_{\kk} X\), the equalities~\eqref{eq:trace-reject} hold
for \((\TT,\FF')\) at \(V\) by the induction hypothesis, and
\(t(V)=t(X)/R(X)\) by Lemma~\ref{lem:trace-quotient}.  Moreover,
\(r_i(t(X))=r_i(X)\) for \(i=1,2\) by Theorem~\ref{thm:trace-reject}, so
\(R(t(X))=R(X)\).  The quotient map \(t(X)\twoheadrightarrow t(X)/R(t(X))\)
therefore coincides with the restriction \(\pi_t\colon t(X)\twoheadrightarrow t(V)\)
of \(\pi\), fitting into the commutative square
\[
\begin{tikzcd}[column sep=large]
 t(X)\arrow[r,twoheadrightarrow,"\pi_t"]\arrow[d,hook]
 &t(V)\arrow[d,hook]\\
 X\arrow[r,twoheadrightarrow,"\pi"]&V
\end{tikzcd}
\]
with \(\pi^{-1}(t(V))=t(X)\).  Consequently, any submodule of \(t(V)\) has
the same inverse image in \(X\) under \(\pi_t\) and under \(\pi\).
Applying~\eqref{eq:transport} to \(M=X\) and to \(M=t(X)\) gives
\[
 r'(X)=\pi^{-1}(r'(V)),\qquad
 r'(t(X))=\pi_t^{-1}(r'(t(V))).
\]
Since \(r'(V)=r'(t(V))\) by the induction hypothesis at \(V\), these two
inverse images coincide, so \(r'(X)=r'(t(X))\).

For the second equality, the isomorphism \(X/r'(X)\cong V/r'(V)\) from~\eqref{eq:transport} carries
\(t(X)/r'(X)\) onto \(t(V)/r'(V)\), and \(t(V/r'(V))=t(V)/r'(V)\) by the
induction hypothesis at \(V\).  Hence \(t(X/r'(X))=t(X)/r'(X)\), completing
the induction on \(\dim_{\kk} X\).  By Theorem~\ref{thm:trace-reject},
\((\TT,\FF_1\cap\FF_2)\) is a pretorsion theory, and induction on \(|I|\)
extends this to any finite index set \(I\).

\medskip
\noindent\textbf{Step 2: Arbitrary intersections.}
Put \(\FF=\bigcap_{i\in I}\FF_i\), and for each finite subset
\(J\subseteq I\) put \(\FF_J=\bigcap_{j\in J}\FF_j\) and \(r_J=r_{\FF_J}\).
By the finite case, \((\TT,\FF_J)\) is a
pretorsion theory for every finite \(J\subseteq I\).  Fix a module \(X\).
Since \(r_J(X)\subseteq r_{J'}(X)\) whenever \(J\subseteq J'\) and \(X\) is
finite-dimensional, we may choose a finite subset \(J_0\subseteq I\) for
which \(\dim_{\kk} r_{J_0}(X)\) is maximal, and put \(N=r_{J_0}(X)\).  For
every \(i\in I\), maximality gives \(r_{J_0\cup\{i\}}(X)=N\), so
\(X/N\in\FF_{J_0\cup\{i\}}\subseteq\FF_i\) by
Proposition~\ref{prop:trace-reject-properties}(2).  Thus \(X/N\in\FF\),
which gives \(r_{\FF}(X)\subseteq N\); since \(\FF\subseteq\FF_{J_0}\), we
also have \(N=r_{J_0}(X)\subseteq r_{\FF}(X)\), and hence
\(r_{\FF}(X)=r_J(X)\) for every finite \(J\supseteq J_0\).  Choosing such a
finite subset for \(X\) and another for \(t(X)\), and taking their union
\(J\), we obtain \(r_{\FF}(X)=r_J(X)\) and \(r_{\FF}(t(X))=r_J(t(X))\).
The equalities~\eqref{eq:trace-reject} for \((\TT,\FF_J)\) at \(X\)
therefore imply those for \((\TT,\FF)\) at \(X\), and
Theorem~\ref{thm:trace-reject} shows that \((\TT,\FF)\) is a pretorsion
theory.
\end{proof}

To construct meets in \(\PT(\Lambda)\), we must also compare pretorsion
theories with different first components.  The next lemma shows that the
first equality in~\eqref{eq:trace-reject}, combined with the second
reconstruction formula in~\eqref{eq:reconstruction}, suffices for a pair
\((\TT,\FF)\) to be a pretorsion theory.

\begin{lemma}\label{lem:one-sided}
Let \(\TT\in\LQ(\Lambda)\) and \(\FF\in\LS(\Lambda)\), put \(\ZZZ=\TT\cap\FF\), and write
\(t=t_{\TT}\) and \(r=r_{\FF}\).  Suppose that \(r(X)=r(t(X))\) for every
module \(X\) and that
\(\TT=\{X\in\mod\Lambda:X/r(X)\in\ZZZ\}\).  Then \((\TT,\FF)\) is a pretorsion
theory.
\end{lemma}

\begin{proof}
Fix \(X\in\mod\Lambda\), put \(U=t(X/r(X))\), and let \(P\subseteq X\) be
the inverse image of \(U\) under the quotient map \(X\twoheadrightarrow X/r(X)\).
Since \(r(X)=r(t(X))\subseteq t(X)\), the submodule \(t(X)/r(X)\) of
\(X/r(X)\) belongs to \(\TT\) and is therefore contained in \(U\), so
\(t(X)\subseteq P\).  Moreover, \(U\) belongs to \(\TT\) and is a submodule
of \(X/r(X)\in\FF\), so \(U\in\ZZZ\).  By definition of \(P\), we have a
commutative diagram with exact rows:
\[
\begin{tikzcd}
 0\arrow[r]&r(X)\arrow[r]\arrow[d,equal]&P\arrow[r]\arrow[d,hook]
   &U\arrow[r]\arrow[d,hook]&0\\
 0\arrow[r]&r(X)\arrow[r]&X\arrow[r]&X/r(X)\arrow[r]&0.
\end{tikzcd}
\]
Since \(t(X)\subseteq P\subseteq X\),
Proposition~\ref{prop:trace-reject-properties}(5) gives
\(r(t(X))\subseteq r(P)\subseteq r(X)\), and hence \(r(P)=r(X)\) because
\(r(t(X))=r(X)\).  Consequently,
\(P/r(P)=P/r(X)=U\in\ZZZ\), so the hypothesis on \(\TT\) gives \(P\in\TT\)
and therefore \(P\subseteq t(X)\).  Thus \(P=t(X)\) and
\(t(X/r(X))=U=t(X)/r(X)\), so \((\TT,\FF)\) is a pretorsion theory by
Theorem~\ref{thm:trace-reject}.
\end{proof}

To apply Lemma~\ref{lem:one-sided} to an intersection of first components,
we must recover membership in each component from a quotient by the reject
of a larger second component.  The following lemma gives this implication.

\begin{lemma}\label{lem:trace-trapping}
Let \((\TT,\FF)\) be a pretorsion theory, let \(\GG\in\LS(\Lambda)\) contain
\(\FF\), and let \(X\) be a module with \(X/r_{\GG}(X)\in\TT\).  Then
\(X\in\TT\).
\end{lemma}

\begin{proof}
Since \(\FF\subseteq\GG\), we have \(r_{\GG}(X)\subseteq r_{\FF}(X)\), so
\(X/r_{\FF}(X)\) is a quotient of \(X/r_{\GG}(X)\in\TT\) and therefore
belongs to \(\TT\).  By Theorem~\ref{thm:trace-reject},
\(X/r_{\FF}(X)=t_{\TT}(X/r_{\FF}(X))=t_{\TT}(X)/r_{\FF}(X)\), which gives
\(t_{\TT}(X)=X\) and hence \(X\in\TT\) by
Proposition~\ref{prop:trace-reject-properties}(3).
\end{proof}

Combining Lemmas~\ref{lem:one-sided} and~\ref{lem:trace-trapping}, we show
that a lower bound \((\TT'',\FF'')\) of a nonempty family
\((\TT_i,\FF_i)_{i\in I}\) yields a
pretorsion theory \((\bigcap_{i\in I}\TT_i,\FF'')\).

\begin{proposition}\label{prop:lower-bound}
Let \((\TT_i,\FF_i)\), \(i\in I\), be a nonempty family in \(\PT(\Lambda)\), and let
\((\TT'',\FF'')\in\PT(\Lambda)\) be a lower bound of the family.  Then
\((\bigcap_{i\in I}\TT_i,\FF'')\) is a pretorsion theory.
\end{proposition}

\begin{proof}
Put \(\widehat{\TT}=\bigcap_{i\in I}\TT_i\in\LQ(\Lambda)\),
\(\widehat{\ZZZ}=\widehat{\TT}\cap\FF''\), and \(r=r_{\FF''}\).  We verify
the two hypotheses of Lemma~\ref{lem:one-sided} for
\((\widehat{\TT},\FF'')\).  First, since \(\TT''\subseteq\widehat{\TT}\),
every morphism from an object of \(\TT''\) to a module \(X\) has image
contained in \(t_{\widehat{\TT}}(X)\), so
\(t_{\TT''}(t_{\widehat{\TT}}(X))=t_{\TT''}(X)\).  Applying
Theorem~\ref{thm:trace-reject} for \((\TT'',\FF'')\) to \(X\) and to
\(t_{\widehat{\TT}}(X)\) yields
\[
 r(X)=r\bigl(t_{\TT''}(X)\bigr)
     =r\bigl(t_{\TT''}(t_{\widehat{\TT}}(X))\bigr)
     =r\bigl(t_{\widehat{\TT}}(X)\bigr).
\]
Second, if \(X\in\widehat{\TT}\), then \(X/r(X)\) belongs to
\(\widehat{\TT}\) by quotient closure and to \(\FF''\) by
Proposition~\ref{prop:trace-reject-properties}(2), so
\(X/r(X)\in\widehat{\ZZZ}\).  Conversely, if \(X/r(X)\in\widehat{\ZZZ}\),
then \(X/r(X)\in\TT_i\) and \(\FF_i\subseteq\FF''\) for every \(i\in I\), so
Lemma~\ref{lem:trace-trapping} gives \(X\in\TT_i\) for every \(i\in I\), that
is, \(X\in\widehat{\TT}\).  Thus
\(\widehat{\TT}=\{X\in\mod\Lambda:X/r(X)\in\widehat{\ZZZ}\}\), and
Lemma~\ref{lem:one-sided} shows that \((\widehat{\TT},\FF'')\) is a
pretorsion theory.
\end{proof}

Propositions~\ref{prop:partner-intersection} and~\ref{prop:lower-bound}
together yield the complete lattice structure on \(\PT(\Lambda)\).

\begin{theorem}\label{thm:PT-lattice}
For every finite-dimensional algebra \(\Lambda\), the poset \(\PT(\Lambda)\) is a complete
lattice.  For a nonempty family \((\TT_i,\FF_i)\), \(i\in I\), its meet is
\begin{equation}\label{eq:PT-meet}
 \Bigl(\,\bigcap_{i\in I}\TT_i,\ \bigcap\{\FF:(\textstyle\bigcap_i\TT_i,\FF)\in
 \PT(\Lambda),\ \FF\supseteq\FF_i\text{ for every }i\}\Bigr).
\end{equation}
Its join is
\begin{equation}\label{eq:PT-join}
 \Bigl(\,\bigcap\{\TT:(\TT,\textstyle\bigcap_i\FF_i)\in\PT(\Lambda),\
 \TT\supseteq\TT_i\text{ for every }i\},\ \bigcap_{i\in I}\FF_i\Bigr).
\end{equation}
The empty meet is \((\mod\Lambda,0)\), and the empty join is
\((0,\mod\Lambda)\).
\end{theorem}

\begin{proof}
Put \(\widehat{\TT}=\bigcap_{i\in I}\TT_i\).  The set of second components in
\eqref{eq:PT-meet} is nonempty because it contains \(\mod\Lambda\), so
Proposition~\ref{prop:partner-intersection} shows that
\((\widehat{\TT},\widehat{\FF})\in\PT(\Lambda)\) for their intersection
\(\widehat{\FF}\).  By construction, \((\widehat{\TT},\widehat{\FF})\) is a
lower bound of the family \((\TT_i,\FF_i)_{i\in I}\).  If
\((\TT'',\FF'')\in\PT(\Lambda)\) is any lower bound of the family, then
\(\TT''\subseteq\widehat{\TT}\), and Proposition~\ref{prop:lower-bound}
shows that \((\widehat{\TT},\FF'')\in\PT(\Lambda)\).  Since
\(\FF''\supseteq\FF_i\) for every \(i\in I\), the subcategory \(\FF''\)
belongs to the family intersected in~\eqref{eq:PT-meet}, so
\(\FF''\supseteq\widehat{\FF}\) and therefore
\((\TT'',\FF'')\leq(\widehat{\TT},\widehat{\FF})\).  Thus every nonempty
family in \(\PT(\Lambda)\) has a meet given by~\eqref{eq:PT-meet}.
Applying formula~\eqref{eq:PT-meet} to \(\Lambda^{\op}\) and using
Remark~\ref{rem:duality} gives~\eqref{eq:PT-join}.  The least and greatest
elements give the empty join and meet, respectively.
\end{proof}

We will use the following criterion both to construct meets with a fixed
intersection and to realize image-closed subcategories in
Theorem~\ref{thm:image-middle}.

\begin{lemma}\label{lem:realization-criterion}
Let \(\TT\in\LQ(\Lambda)\), \(\FF\in\LS(\Lambda)\),
\(\ZZZ=\TT\cap\FF\), and \(t=t_{\TT}\).  Suppose that
\begin{align}
 \FF&=\{Y\in\mod\Lambda:t(Y)\in\ZZZ\},\label{eq:realization-F}\\
 \TT&=\{E\in\mod\Lambda:\text{every morphism }E\to F,
       \ F\in\FF,\text{ is }\ZZZ\text{-trivial}\}.
       \label{eq:realization-T}
\end{align}
Then \((\TT,\FF)\) is a pretorsion theory.
\end{lemma}

\begin{proof}
For \(X\in\mod\Lambda\), put \(K=r_{\ZZZ}(t(X))\).  By
Lemma~\ref{lem:finite-intersection}, the quotient \(t(X)/K\) embeds in a
finite direct sum of objects of \(\ZZZ\), so \(t(X)/K\in\FF\) since
\(\ZZZ\subseteq\FF\) and \(\FF\) is submodule-closed; since \(t(X)/K\) is
also a quotient of \(t(X)\in\TT\), we have \(t(X)/K\in\TT\cap\FF=\ZZZ\).
Put \(U=t(X/K)\) and let \(E\) be the inverse image of \(U\) in \(X\).
Since \(t(X)/K\in\TT\), we have \(t(X)/K\subseteq U\) and hence
\(t(X)\subseteq E\).  We obtain a commutative diagram with exact rows:
\[
\begin{tikzcd}
0\arrow[r]&K\arrow[r]\arrow[d,equal]&E\arrow[r]\arrow[d,hook]
 &U\arrow[r]\arrow[d,hook]&0\\
0\arrow[r]&K\arrow[r]&X\arrow[r]&X/K\arrow[r]&0.
\end{tikzcd}
\]
By~\eqref{eq:realization-F}, every morphism from \(t(X)\) to an object
\(F\in\FF\) factors through \(t(F)\in\ZZZ\) and therefore vanishes on \(K\).
In particular, every morphism \(E\to F\) with \(F\in\FF\) vanishes on
\(K\subseteq t(X)\) and thus descends to a morphism \(U\to F\), which is
\(\ZZZ\)-trivial by~\eqref{eq:realization-T} because \(U\in\TT\).  Hence
every morphism \(E\to F\) is \(\ZZZ\)-trivial, so~\eqref{eq:realization-T}
gives \(E\in\TT\).  Consequently, \(E=t(X)\) and
\begin{equation}\label{eq:realization-trace}
                         t(X/K)=t(X)/K\in\ZZZ,
\end{equation}
which implies \(X/K\in\FF\) by~\eqref{eq:realization-F}.

Since every morphism from \(t(X)\) or \(X\) to an object \(F\in\FF\)
vanishes on \(K\), we have \(K\subseteq r_{\FF}(t(X))\) and
\(K\subseteq r_{\FF}(X)\).  Conversely, since
\(t(X)/K\in\ZZZ\subseteq\FF\) and \(X/K\in\FF\), we obtain the reverse
inclusions.  Therefore \(r_{\FF}(t(X))=K=r_{\FF}(X)\),
and~\eqref{eq:realization-trace} then gives both equalities of
Theorem~\ref{thm:trace-reject}.
\end{proof}

The lattice operations also preserve a prescribed intersection.  For a
subcategory \(\ZZZ\), write
\[
 \PT_{\ZZZ}(\Lambda)=\{(\TT,\FF)\in\PT(\Lambda):\TT\cap\FF=\ZZZ\}.
\]
Theorem~\ref{thm:image-middle} shows that this set is nonempty precisely when
\(\ZZZ\) is image-closed.

\begin{proposition}\label{prop:fixed-intersection-lattice}
For every subcategory \(\ZZZ\), the set \(\PT_{\ZZZ}(\Lambda)\) is closed
under all nonempty meets and joins in \(\PT(\Lambda)\).
\end{proposition}

\begin{proof}
Let \((\TT_i,\FF_i)_{i\in I}\) be a nonempty family in
\(\PT_{\ZZZ}(\Lambda)\).  Put \(\TT=\bigcap_{i\in I}\TT_i\), \(t=t_{\TT}\),
and
\[
 \FF=\{Y\in\mod\Lambda:t(Y)\in\ZZZ\}.
\]
We first show that \((\TT,\FF)\) is a pretorsion theory with intersection
\(\ZZZ\), and then compare it with the meet in \(\PT(\Lambda)\).  We
repeatedly use the following observation: if a module \(M\in\TT\) embeds in an
object of \(\ZZZ\), then for any \(i\in I\) we have \(M\in\TT_i\) and
\(M\in\FF_i\) (because \(\ZZZ\subseteq\FF_i\) and \(\FF_i\) is
submodule-closed), so \(M\in\TT_i\cap\FF_i=\ZZZ\).

\medskip
\noindent\textbf{Step 1: Constructing a pretorsion theory with intersection \(\ZZZ\).}
If \(Y'\subseteq Y\in\FF\), then \(t(Y')\subseteq t(Y)\in\ZZZ\), so the
observation gives \(t(Y')\in\ZZZ\).  Thus \(\FF\) is submodule-closed;
additivity follows from that of \(t\).  We have \(\ZZZ\subseteq\TT\cap\FF\),
and \(X\in\TT\cap\FF\) implies \(X=t(X)\in\ZZZ\), so
\(\TT\cap\FF=\ZZZ\).  If \(Y\in\FF_i\), then
\(t(Y)\subseteq t_{\TT_i}(Y)\in\ZZZ\) by~\eqref{eq:reconstruction}; since
\(t(Y)\in\TT\), the observation gives \(t(Y)\in\ZZZ\) and hence \(Y\in\FF\).
Thus \(\FF_i\subseteq\FF\) for every \(i\in I\).

To apply Lemma~\ref{lem:realization-criterion}, it remains to verify
\eqref{eq:realization-T}.  Every morphism \(T\to Y\) with \(T\in\TT\) and
\(Y\in\FF\) factors through \(t(Y)\in\ZZZ\).  Conversely, suppose that every morphism from a module \(E\)
to an object of \(\FF\) factors through \(\ZZZ\).  For each \(i\in I\), we
have \(E/r_{\FF_i}(E)\in\FF_i\subseteq\FF\), so the canonical projection
\(p_i\colon E\twoheadrightarrow E/r_{\FF_i}(E)\) is \(\ZZZ\)-trivial.  By
Theorem~\ref{thm:trace-reject}, the inclusion
\(t_{\TT_i}(E)\hookrightarrow E\) is the \(\ZZZ\)-prekernel of \(p_i\); the
identity \(1_E\) therefore factors through \(t_{\TT_i}(E)\), giving
\(E\in\TT_i\).  Since \(i\in I\) was arbitrary, \(E\in\TT\).

This proves~\eqref{eq:realization-T}, so
Lemma~\ref{lem:realization-criterion} gives \((\TT,\FF)\in\PT(\Lambda)\).

\medskip
\noindent\textbf{Step 2: The ambient meet and join.}
By~\eqref{eq:PT-meet}, the meet in \(\PT(\Lambda)\) has the form
\((\TT,\GG)\), where \(\FF_i\subseteq\GG\subseteq\FF\) for every \(i\in I\).
Fixing \(i\in I\), we obtain
\[
 \ZZZ=\TT\cap\FF_i\subseteq\TT\cap\GG\subseteq\TT\cap\FF=\ZZZ,
\]
so \(\TT\cap\GG=\ZZZ\) and the meet belongs to \(\PT_{\ZZZ}(\Lambda)\).  The
assertion for nonempty joins follows by duality
(Remark~\ref{rem:duality}).
\end{proof}

When \(\ZZZ=0\), Proposition~\ref{prop:fixed-intersection-lattice} shows that
torsion pairs form a complete sublattice of \(\PT(\Lambda)\).

\begin{corollary}\label{cor:torsion-sublattice}
The torsion pairs form a complete sublattice containing the least and
greatest elements of \(\PT(\Lambda)\).
Sending a torsion pair to its first component identifies this sublattice
with the lattice of torsion classes ordered by inclusion.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:zero-middle-torsion}, the torsion pairs in
\(\mod\Lambda\) are the elements of \(\PT_0(\Lambda)\).
Proposition~\ref{prop:fixed-intersection-lattice} gives closure under
nonempty meets and joins, and the greatest and least elements
\((\mod\Lambda,0)\) and \((0,\mod\Lambda)\) of \(\PT(\Lambda)\) also belong
to \(\PT_0(\Lambda)\).  Finally, the reconstruction formula~\eqref{eq:reconstruction} at
\(\ZZZ=0\) gives
\(\FF=\{Y\in\mod\Lambda:\Hom_\Lambda(\TT,Y)=0\}\).
Thus \(\TT\) determines \(\FF\), and
\(\TT_1\subseteq\TT_2\) implies \(\FF_1\supseteq\FF_2\), so the projection
\((\TT,\FF)\mapsto\TT\) is an order isomorphism.
\end{proof}
\section{Image-closed subcategories as intersections of the two components}
\label{sec:middles}

In this section, we prove Theorem~\ref{thm:intro-main} by characterizing the
subcategories of \(\mod\Lambda\) that occur as intersections of pretorsion
theories, and we construct the greatest and least pretorsion theories with a
prescribed intersection.  We call a pretorsion theory \((\TT,\FF)\) with
\(\TT\cap\FF=\ZZZ\) a \emph{realization} of \(\ZZZ\).

\begin{definition}\label{def:gen-cogen}
For a subcategory \(\ZZZ\subseteq\mod \Lambda\), let \(\Gen\ZZZ\)
consist of the quotients of finite direct sums of objects of \(\ZZZ\), and
let \(\Cogen\ZZZ\) consist of the submodules of such direct sums.
\end{definition}

For an image-closed subcategory \(\ZZZ\), Theorem~\ref{thm:image-middle}
below shows that the quotient-closed subcategory \(\Gen\ZZZ\) and the
submodule-closed subcategory \(\Cogen\ZZZ\) have intersection \(\ZZZ\), but
the pair \((\Gen\ZZZ,\Cogen\ZZZ)\) need not itself be a pretorsion theory.
\begin{example}\label{ex:semisimple-middle}
Let \(\Lambda_2=\kk(1\to 2)\) and let \(\ZZZ\) be the
subcategory of semisimple \(\Lambda_2\)-modules.  Then
\(\Gen\ZZZ=\Cogen\ZZZ=\ZZZ\).  Writing \(t=t_{\ZZZ}\) and \(r=r_{\ZZZ}\), we
see that for the indecomposable representation \(X=(\kk\xrightarrow{1}\kk)\),
both \(t(X)\) and \(r(X)\) are equal to the simple submodule
\(S_2=(0\to\kk)\), whereas \(r(S_2)=0\) because \(S_2\in\ZZZ\).  Thus
\(r(X)\neq r(t(X))=0\), so \((\ZZZ,\ZZZ)\) is not a pretorsion theory by
Theorem~\ref{thm:trace-reject}.
\end{example}

To obtain a pretorsion theory with intersection \(\ZZZ\), we enlarge the
first component.  If \((\TT,\FF)\) is a pretorsion theory with
\(\TT\cap\FF=\ZZZ\), then for any morphism \(f\colon X\to N\) with
\(X\in\TT\) and \(N\in\ZZZ\subseteq\FF\), the image \(\Image f\) lies in
\(\TT\) by quotient closure and in \(\FF\) by submodule closure, and hence
belongs to \(\ZZZ\).  This necessary condition on \(\TT\), together with the
reconstruction formula~\eqref{eq:reconstruction} for \(\FF\), motivates the
following construction.

\begin{construction}\label{def:middle-endpoints}
For an image-closed subcategory \(\ZZZ\subseteq\mod \Lambda\) put
\begin{align*}
 \TT^+(\ZZZ)&=\{X\in\mod\Lambda:\Image f\in\ZZZ\text{ for every }
                f\colon X\to N\text{ with }N\in\ZZZ\},\\
 \FF^-(\ZZZ)&=\{Y\in\mod\Lambda:t_{\TT^+(\ZZZ)}(Y)\in\ZZZ\}.
\end{align*}
\end{construction}

The superscripts anticipate the extremality proved in
Corollary~\ref{cor:greatest-realization}: among realizations of \(\ZZZ\),
\(\TT^+(\ZZZ)\) is the largest possible first component and
\(\FF^-(\ZZZ)\) the smallest possible second component.

The equivalence of (1) and (2) in the following theorem is due to Auslander
and Smal{\o} \cite[Proposition~4.10(i)]{AuslanderSmalo}.  We include its
short proof before showing that \((\TT^+(\ZZZ),\FF^-(\ZZZ))\) is a
pretorsion theory with intersection \(\ZZZ\).

\begin{theorem}\label{thm:image-middle}
For a subcategory \(\ZZZ\subseteq\mod \Lambda\), the following are
equivalent.
\begin{enumerate}
\item \(\ZZZ\) is image-closed.
\item \(\ZZZ=\Gen\ZZZ\cap\Cogen\ZZZ\).
\item \(\ZZZ=\TT\cap\FF\) for a pretorsion theory \((\TT,\FF)\).
\end{enumerate}
\end{theorem}

\begin{proof}
(1) \(\Rightarrow\) (2): Let \(X\in\Gen\ZZZ\cap\Cogen\ZZZ\).  By
Definition~\ref{def:gen-cogen}, there exist finite direct sums \(N_1\) and
\(N_2\) of objects of \(\ZZZ\), an epimorphism \(N_1\twoheadrightarrow X\),
and a monomorphism \(X\hookrightarrow N_2\):
\[
                         N_1\twoheadrightarrow X\lhook\joinrel\longrightarrow N_2.
\]
Since \(\ZZZ\) is additive, both \(N_1\) and \(N_2\) belong to \(\ZZZ\).
The composite morphism \(N_1\to N_2\) has image \(X\), so \(X\in\ZZZ\)
because \(\ZZZ\) is image-closed.  The reverse inclusion
\(\ZZZ\subseteq\Gen\ZZZ\cap\Cogen\ZZZ\) is immediate.

(2) \(\Rightarrow\) (1): For any morphism \(f\colon X\to Y\) with
\(X,Y\in\ZZZ\), the image \(\Image f\) is a quotient of \(X\) and a
submodule of \(Y\), so \(\Image f\in\Gen\ZZZ\cap\Cogen\ZZZ=\ZZZ\).

(3) \(\Rightarrow\) (1): If \((\TT,\FF)\) is a pretorsion theory with
\(\TT\cap\FF=\ZZZ\), then \(\TT\in\LQ(\Lambda)\) and \(\FF\in\LS(\Lambda)\)
by Proposition~\ref{prop:pretorsion-closure}, so \(\ZZZ=\TT\cap\FF\) is
image-closed by~\eqref{eq:image-in-intersection}.

(1) \(\Rightarrow\) (3): Assume that \(\ZZZ\) is image-closed, and consider
the pair \((\TT^+(\ZZZ),\FF^-(\ZZZ))\) of
Construction~\ref{def:middle-endpoints}.  We first verify the closure
properties of the two components and show that their intersection is \(\ZZZ\),
and then apply Lemma~\ref{lem:realization-criterion}.

\medskip
\noindent\textbf{Step 1: Closure properties and the intersection.}
Write \(t=t_{\TT^+(\ZZZ)}\).  We first verify that
\(\TT^+(\ZZZ)\in\LQ(\Lambda)\), \(\FF^-(\ZZZ)\in\LS(\Lambda)\), and both
subcategories contain \(\ZZZ\).

Let \(X_1,X_2\in\TT^+(\ZZZ)\) and let \(f\colon X_1\oplus X_2\to N\) be a
morphism with \(N\in\ZZZ\).  The images of the two components
\(f_i\colon X_i\to N\) (\(i=1,2\)) belong to \(\ZZZ\), and factoring each
\(f_i\) through \(\Image f_i\) yields a commutative triangle
\[
\begin{tikzcd}[column sep=large]
 X_1\oplus X_2\arrow[r,twoheadrightarrow]\arrow[dr,swap,"f"]
   & \Image f_1\oplus\Image f_2\arrow[d,"{(\iota_1,\iota_2)}"]\\
   & N,
\end{tikzcd}
\]
where \(\iota_i\colon\Image f_i\hookrightarrow N\) is the inclusion.  The
vertical morphism has source \(\Image f_1\oplus\Image f_2\in\ZZZ\), target
\(N\in\ZZZ\), and image \(\Image f\), so \(\Image f\in\ZZZ\) because
\(\ZZZ\) is image-closed.  Thus \(X_1\oplus X_2\in\TT^+(\ZZZ)\).  Clearly
\(0\in\TT^+(\ZZZ)\), and \(\TT^+(\ZZZ)\) is closed under isomorphisms.  If \(X\in\TT^+(\ZZZ)\) and
\(\pi\colon X\twoheadrightarrow X/K\) is a quotient map, then any morphism
\(f\colon X/K\to N\) with \(N\in\ZZZ\) satisfies
\(\Image f=\Image(f\pi)\in\ZZZ\).  Thus \(\TT^+(\ZZZ)\) is
quotient-closed and in particular closed under direct summands, so
\(\TT^+(\ZZZ)\in\LQ(\Lambda)\).

Next, we record the identity
\begin{equation}\label{eq:Tplus-intersection}
                     \TT^+(\ZZZ)\cap\Cogen\ZZZ=\ZZZ.
\end{equation}
Indeed, if \(X\in\TT^+(\ZZZ)\cap\Cogen\ZZZ\), there is a monomorphism
\(X\hookrightarrow N\) into a finite direct sum \(N\) of objects of
\(\ZZZ\); since \(N\in\ZZZ\) and \(X\in\TT^+(\ZZZ)\), the image \(X\) of
this monomorphism belongs to \(\ZZZ\).  Conversely, image closure of
\(\ZZZ\) implies \(\ZZZ\subseteq\TT^+(\ZZZ)\), and clearly
\(\ZZZ\subseteq\Cogen\ZZZ\).

We now check that \(\FF^-(\ZZZ)\in\LS(\Lambda)\).  Since \(t\) commutes with
finite direct sums by Proposition~\ref{prop:trace-reject-properties}(4) and
\(\ZZZ\) is additive and closed under direct summands, \(\FF^-(\ZZZ)\) is
additive and closed under direct summands.  If \(W\subseteq Y\) with
\(Y\in\FF^-(\ZZZ)\), then \(t(W)\subseteq t(Y)\in\ZZZ\) by
Proposition~\ref{prop:trace-reject-properties}(5), so \(t(W)\in\Cogen\ZZZ\);
since \(t(W)\in\TT^+(\ZZZ)\) by
Proposition~\ref{prop:trace-reject-properties}(1), equation~\eqref{eq:Tplus-intersection}
gives \(t(W)\in\ZZZ\), and thus \(W\in\FF^-(\ZZZ)\).  Hence
\(\FF^-(\ZZZ)\in\LS(\Lambda)\).

Finally, since \(\ZZZ\subseteq\TT^+(\ZZZ)\), every \(N\in\ZZZ\) satisfies
\(t(N)=N\in\ZZZ\) by Proposition~\ref{prop:trace-reject-properties}(3), so
\(\ZZZ\subseteq\FF^-(\ZZZ)\) as well.  Conversely, if
\(X\in\TT^+(\ZZZ)\cap\FF^-(\ZZZ)\), then \(X=t(X)\in\ZZZ\), so
\(\TT^+(\ZZZ)\cap\FF^-(\ZZZ)=\ZZZ\).

\medskip
\noindent\textbf{Step 2: The pretorsion property.}
The definition of \(\FF^-(\ZZZ)\) gives~\eqref{eq:realization-F}.
For~\eqref{eq:realization-T}, every morphism from an object of
\(\TT^+(\ZZZ)\) to \(Y\in\FF^-(\ZZZ)\) factors through \(t(Y)\in\ZZZ\).
Conversely, suppose that every morphism from \(E\) to an object of
\(\FF^-(\ZZZ)\) is \(\ZZZ\)-trivial.  If \(Y\in\Cogen\ZZZ\), then
\(t(Y)\in\TT^+(\ZZZ)\cap\Cogen\ZZZ=\ZZZ\) by~\eqref{eq:Tplus-intersection},
so \(\Cogen\ZZZ\subseteq\FF^-(\ZZZ)\).  For a morphism \(f\colon E\to N\)
with \(N\in\ZZZ\), put \(I=\Image f\).  Since
\(I\in\Cogen\ZZZ\subseteq\FF^-(\ZZZ)\), the epimorphism
\(E\twoheadrightarrow I\) factors through an object of \(\ZZZ\), giving
\(I\in\Gen\ZZZ\).  Thus \(I\in\Gen\ZZZ\cap\Cogen\ZZZ=\ZZZ\) by (2),
and \(E\in\TT^+(\ZZZ)\).  Lemma~\ref{lem:realization-criterion} now shows
that \((\TT^+(\ZZZ),\FF^-(\ZZZ))\) is a pretorsion theory.
\end{proof}

The realizing pair \((\TT^+(\ZZZ),\FF^-(\ZZZ))\) has the following extremal
property in \(\PT(\Lambda)\).

\begin{corollary}\label{cor:greatest-realization}
Let \(\ZZZ\) be image-closed.  Then
\((\TT^+(\ZZZ),\FF^-(\ZZZ))\) is the greatest pretorsion theory with
intersection \(\ZZZ\), in the order~\eqref{eq:PT-order}.
\end{corollary}

\begin{proof}
Let \((\TT,\FF)\in\PT(\Lambda)\) satisfy \(\TT\cap\FF=\ZZZ\).
For \(X\in\TT\) and \(N\in\ZZZ\), the image of every map \(X\to N\)
belongs to \(\TT\cap\FF=\ZZZ\).  Hence
\(\TT\subseteq\TT^+(\ZZZ)\) by Construction~\ref{def:middle-endpoints}.
If \(Y\in\FF^-(\ZZZ)\), then
\(t_{\TT}(Y)\subseteq t_{\TT^+(\ZZZ)}(Y)\in\ZZZ\subseteq\FF\).  Since
\(\FF\) is submodule-closed, we obtain \(t_{\TT}(Y)\in\TT\cap\FF=\ZZZ\), and
Corollary~\ref{cor:reconstruction} gives \(Y\in\FF\).  Thus
\(\FF^-(\ZZZ)\subseteq\FF\), so
\((\TT,\FF)\leq(\TT^+(\ZZZ),\FF^-(\ZZZ))\).
\end{proof}

For example, when \(\ZZZ\) is the subcategory of semisimple
\(\Lambda_2\)-modules in Example~\ref{ex:semisimple-middle},
every morphism to a semisimple module has semisimple image, so
\(\TT^+(\ZZZ)=\mod\Lambda_2\) and \(\FF^-(\ZZZ)=\ZZZ\); thus
\((\mod\Lambda_2,\ZZZ)\) is the greatest realization of \(\ZZZ\).

The dual construction yields the least realization.

\begin{construction}\label{def:dual-middle-endpoints}
For an image-closed subcategory \(\ZZZ\subseteq\mod\Lambda\), put
\begin{align*}
 \FF^+(\ZZZ)&=\{Y\in\mod\Lambda:\Image g\in\ZZZ\text{ for every }
                   g\colon N\to Y\text{ with }N\in\ZZZ\},\\
 \TT^-(\ZZZ)&=\{X\in\mod\Lambda:X/r_{\FF^+(\ZZZ)}(X)\in\ZZZ\}.
\end{align*}
\end{construction}

\begin{corollary}\label{cor:least-realization}
Let \(\ZZZ\) be image-closed.  Then \((\TT^-(\ZZZ),\FF^+(\ZZZ))\) is the
least pretorsion theory with intersection \(\ZZZ\).  Every pretorsion
theory \((\TT,\FF)\) with \(\TT\cap\FF=\ZZZ\) satisfies
\[
 (\TT^-(\ZZZ),\FF^+(\ZZZ))\leq(\TT,\FF)
 \leq(\TT^+(\ZZZ),\FF^-(\ZZZ)).
\]
Moreover, \(\PT_{\ZZZ}(\Lambda)\) is a complete lattice: its nonempty
meets and joins agree with those in \(\PT(\Lambda)\), and its empty meet
and join are the greatest and least realizations above.
\end{corollary}

\begin{proof}
Applying Corollary~\ref{cor:greatest-realization} to the image-closed
subcategory \(\DDual\ZZZ\subseteq\mod\Lambda^{\op}\) and using
Remark~\ref{rem:duality} shows that \((\TT^-(\ZZZ),\FF^+(\ZZZ))\) is the
least realization of \(\ZZZ\).  Proposition~\ref{prop:fixed-intersection-lattice}
gives closure under nonempty meets and joins in \(\PT(\Lambda)\), while the
greatest and least realizations provide the empty meet and join.
\end{proof}

\subsection{The convex geometry of image closure}\label{subsec:convex}

We now study \(\LI(\Lambda)\) as a complete lattice and use convex geometry
to describe its generating sets and covering relations.  While meets in
\(\LI(\Lambda)\) are intersections, forming joins requires the smallest
image-closed subcategory containing a given family, described by the image
closure of Auslander and Smal{\o} \cite[Proposition~4.3]{AuslanderSmalo}.

\begin{definition}\label{def:image-closure}
For a subcategory \(\CC\subseteq\mod \Lambda\), its \emph{image closure} is
\[
                         \icl\CC=\Gen\CC\cap\Cogen\CC .
\]
\end{definition}

\begin{corollary}[{\cite[Proposition~4.3]{AuslanderSmalo}}]
\label{cor:image-closure}
Let \(\CC\) be a subcategory of \(\mod\Lambda\).  Then \(\icl\CC\) is the smallest
image-closed subcategory of \(\mod \Lambda\) containing \(\CC\).  In particular a
subcategory is image-closed if and only if it equals its own image
closure.
\end{corollary}

\begin{proof}
Since \(\Gen\CC\) is quotient-closed and \(\Cogen\CC\) is submodule-closed,
their intersection \(\icl\CC\) is image-closed by~\eqref{eq:image-in-intersection}, and clearly
\(\CC\subseteq\icl\CC\).  If \(\DD\in\LI(\Lambda)\) contains \(\CC\), then
\(\Gen\CC\subseteq\Gen\DD\) and \(\Cogen\CC\subseteq\Cogen\DD\), so the
equivalence of (1) and (2) in Theorem~\ref{thm:image-middle} gives
\[
 \icl\CC\subseteq\Gen\DD\cap\Cogen\DD=\DD .
\]
Thus \(\icl\CC\) is the smallest image-closed subcategory containing \(\CC\),
and \(\CC\in\LI(\Lambda)\) if and only if \(\CC=\icl\CC\).
\end{proof}

The theory of convex geometries on finite sets goes back to Edelman and
Jamison \cite{EdelmanJamison}; we use the following finitary formulation.

\begin{definition}[{\cite[Definitions~2.1 and~2.2]{EnomotoEquidistribution}}]
\label{def:convex-geometry}
A \emph{closure operator} on a set \(E\) is a map \(c\colon 2^E\to 2^E\)
such that \(S\subseteq c(S)\), \(c(S)\subseteq c(S')\) whenever
\(S\subseteq S'\), and \(c(c(S))=c(S)\) for all \(S,S'\subseteq E\).  A
subset \(C\subseteq E\) is \emph{closed} if \(c(C)=C\).  The operator \(c\)
is \emph{finitary} if \(c(S)\) is the union of \(c(S_0)\) over all finite
subsets \(S_0\subseteq S\), and it satisfies \emph{anti-exchange} if
\[
 x\in c(C\cup\{y\})\quad\Longrightarrow\quad y\notin c(C\cup\{x\})
\]
for every closed subset \(C\subseteq E\) and all distinct
\(x,y\in E\setminus C\).  A \emph{convex geometry} is a pair \((E,c)\) where
\(c\) is a closure operator on \(E\) satisfying \(c(\varnothing)=\varnothing\)
and anti-exchange; it is a \emph{finitary convex geometry} if \(c\) is also
finitary.  For a closed subset \(C\subseteq E\), the set of \emph{extreme
points} of \(C\) is
\[
 \ex(C)=\{x\in C:x\notin c(C\setminus\{x\})\}.
\]
\end{definition}

For \(S\subseteq\ind \Lambda\), put
\begin{equation}\label{eq:three-closures}
 \qcl(S)=\ind\Gen(\add S),\qquad
 \scl(S)=\ind\Cogen(\add S),
\end{equation}
\begin{equation}\label{eq:icl-on-subsets}
 \icl(S)=\ind\bigl(\Gen(\add S)\cap\Cogen(\add S)\bigr)
        =\qcl(S)\cap\scl(S).
\end{equation}
Thus \(\icl(S)=\ind(\icl(\add S))\); we use the same symbol \(\icl\) for the
operator on subcategories and for the induced operator on subsets of
\(\ind\Lambda\).  Under the bijection \(\CC\mapsto\ind\CC\) between
subcategories of \(\mod\Lambda\) and subsets of \(\ind\Lambda\), the closed
sets of \(\qcl\) correspond to \(\LQ(\Lambda)\), since \(\Gen\CC\) is the
smallest quotient-closed subcategory containing \(\CC\).  Dually, the closed
sets of \(\scl\) correspond to \(\LS(\Lambda)\), and the closed sets of
\(\icl\) correspond to \(\LI(\Lambda)\) by
Corollary~\ref{cor:image-closure}.

On a finite ground set, the pointwise intersection of two closure operators
of convex geometries is again the closure operator of a convex geometry, by
Edelman and Jamison \cite[Theorem~5.1]{EdelmanJamison}.  We verify the
finitary version directly from~\eqref{eq:icl-on-subsets}.

\begin{theorem}\label{thm:icl-convex}
Let \(\Lambda\) be a finite-dimensional algebra over a field.  Then
\((\ind \Lambda,\icl)\) is a finitary convex geometry whose closed sets
correspond via \(\CC\mapsto\ind\CC\) to the image-closed subcategories of
\(\mod \Lambda\).
\end{theorem}

\begin{proof}
By Corollary~\ref{cor:image-closure}, \(\icl\) is a closure operator on
\(\ind\Lambda\) whose closed sets correspond to \(\LI(\Lambda)\), and
\(\icl(\varnothing)=\varnothing\) because the zero subcategory is
image-closed.

To see that \(\icl\) is finitary, let \(X\in\icl(S)\).  Then \(X\) is a
quotient of a finite direct sum of modules in \(S\) and a submodule of
another such finite direct sum; if \(S_0\subseteq S\) is the finite set of
modules appearing in these two direct sums, then \(X\in\icl(S_0)\).

For anti-exchange, recall from
\cite[Proposition~2.13 and Corollary~2.14]{EnomotoEquidistribution} that both
\(\qcl\) and \(\scl\) satisfy anti-exchange.  Let \(C\subseteq\ind\Lambda\)
be \(\icl\)-closed, and suppose for contradiction that there exist distinct
\(x,y\in\ind\Lambda\setminus C\) with
\[
 x\in\icl(C\cup\{y\}),\qquad y\in\icl(C\cup\{x\}).
\]
Since \(x\notin C=\qcl(C)\cap\scl(C)\), choose
\(c\in\{\qcl,\scl\}\) with \(x\notin c(C)\).  There are two cases.

\medskip
\noindent\textbf{Case 1: \(y\in c(C)\).}
Then \(c(C\cup\{y\})=c(C)\), contradicting
\(x\in\icl(C\cup\{y\})\subseteq c(C\cup\{y\})\).

\medskip
\noindent\textbf{Case 2: \(y\notin c(C)\).}
The two assumed inclusions imply
\(x\in c(c(C)\cup\{y\})\) and \(y\in c(c(C)\cup\{x\})\), contradicting
anti-exchange for \(c\) at the closed set \(c(C)\).
\end{proof}

To describe the extreme points of \(\icl\), we recall the splitting
projective and splitting injective objects introduced by Auslander and
Smal{\o} \cite[Section~2]{AuslanderSmalo}.

\begin{definition}[{\cite[Section~2]{AuslanderSmalo}}]
\label{def:splitting-projective}
Let \(\CC\subseteq\mod \Lambda\) be a subcategory and let
\(X\in\ind\CC\).  Then \(X\) is \emph{splitting projective} in \(\CC\) if
every epimorphism \(N\to X\) with \(N\in\CC\) splits, and \emph{splitting
injective} in \(\CC\) if every monomorphism \(X\to N\) with \(N\in\CC\)
splits.  Write \(\PZ(\CC)\) and \(\IZ(\CC)\) for the sets of splitting
projective and splitting injective objects in \(\ind\CC\).
\end{definition}

Parts~(1) and~(2) of the next proposition follow from Auslander and
Smal{\o} \cite[Proposition~2.2]{AuslanderSmalo} and apply to arbitrary
subcategories; combining them yields a characterization of the extreme
points of an image-closed subcategory.

\begin{proposition}\label{prop:extreme-points}
Let \(\CC\subseteq\mod \Lambda\) be a subcategory and let
\(X\in\ind\CC\).
\begin{enumerate}
\item \(X\notin\qcl(\ind\CC\setminus\{X\})\) if and only if
      \(X\in\PZ(\CC)\).
\item \(X\notin\scl(\ind\CC\setminus\{X\})\) if and only if
      \(X\in\IZ(\CC)\).
\end{enumerate}
If \(\CC\) is image-closed, so that \(\ind\CC\) is \(\icl\)-closed, the
following are equivalent.
\begin{enumerate}
\setcounter{enumi}{2}
\item \(X\in\ex(\ind\CC)\) with respect to \(\icl\).
\item \(\add(\ind\CC\setminus\{X\})\) is image-closed.
\item \(X\in\PZ(\CC)\cup\IZ(\CC)\).
\end{enumerate}
\end{proposition}

\begin{proof}
(1) We prove both directions by contraposition.  If \(X\notin\PZ(\CC)\),
choose a nonsplit epimorphism \(M\twoheadrightarrow X\) with \(M\in\CC\),
and write \(M\cong X^n\oplus M'\) where \(X\) is not a direct summand of
\(M'\), so that \(M'\in\add(\ind\CC\setminus\{X\})\).  By Auslander and
Smal{\o} \cite[Proposition~2.2(a)]{AuslanderSmalo}, the existence of a
nonsplit epimorphism \(M'\oplus X^n\twoheadrightarrow X\) with \(X\)
indecomposable implies the existence of an epimorphism
\((M')^m\twoheadrightarrow X\) for some \(m\geq 1\), whence
\(X\in\qcl(\ind\CC\setminus\{X\})\).  Conversely, if
\(X\in\qcl(\ind\CC\setminus\{X\})\), there is an epimorphism
\(f\colon M\twoheadrightarrow X\) with
\(M\in\add(\ind\CC\setminus\{X\})\subseteq\CC\).  If \(f\) split, then \(X\)
would be isomorphic to a direct summand of \(M\), contradicting the
Krull--Schmidt theorem; thus \(f\) does not split and \(X\notin\PZ(\CC)\).

(2) This is dual to (1).

(3) \(\Leftrightarrow\) (4): Since \(\CC\) is image-closed, we have
\(\icl(\ind\CC\setminus\{X\})\subseteq\ind\CC\), so
\(\icl(\ind\CC\setminus\{X\})\) equals either \(\ind\CC\setminus\{X\}\) or
\(\ind\CC\).  Thus \(X\notin\icl(\ind\CC\setminus\{X\})\) if and only if
\(\icl(\ind\CC\setminus\{X\})=\ind\CC\setminus\{X\}\), which by
Corollary~\ref{cor:image-closure} means that
\(\add(\ind\CC\setminus\{X\})\) is image-closed.

(3) \(\Leftrightarrow\) (5): By~\eqref{eq:icl-on-subsets}, we have
\[
 \icl(\ind\CC\setminus\{X\})
 =\qcl(\ind\CC\setminus\{X\})\cap\scl(\ind\CC\setminus\{X\}).
\]
Hence \(X\notin\icl(\ind\CC\setminus\{X\})\) if and only if
\(X\notin\qcl(\ind\CC\setminus\{X\})\) or
\(X\notin\scl(\ind\CC\setminus\{X\})\), which is equivalent to
\(X\in\PZ(\CC)\cup\IZ(\CC)\) by (1) and~(2).
\end{proof}

The closed sets of any closure operator form a complete lattice, giving
part~(1) below directly from Corollary~\ref{cor:image-closure}.
The convex geometry of Theorem~\ref{thm:icl-convex} gives parts~(2) and~(3).

\begin{corollary}\label{cor:LI-structure}
Let \(\Lambda\) be a finite-dimensional algebra over a field.
\begin{enumerate}
\item \(\LI(\Lambda)\) is a complete lattice, in which the meet of a family
      \((\ZZZ_i)_{i\in I}\) is its intersection and the join is
      \(\icl\bigl(\add\bigcup_i\ZZZ_i\bigr)\).
\item If \(\ZZZ\) is an image-closed subcategory with finitely many
      indecomposable objects, then there is a unique smallest subset
      \(S\subseteq\ind\ZZZ\) with \(\icl(S)=\ind\ZZZ\), namely
      \(S=\PZ(\ZZZ)\cup\IZ(\ZZZ)\).
\item If \(\ZZZ'\subsetneq\ZZZ\) is a
      covering relation in \(\LI(\Lambda)\), then
      \(\ind\ZZZ\setminus\ind\ZZZ'\) has exactly one element.  If \(\Lambda\) is representation-finite, then
      \(\LI(\Lambda)\) is graded by the size
      \(\ZZZ\mapsto|\ind\ZZZ|\).
\end{enumerate}
\end{corollary}

\begin{proof}
(1) Any intersection of image-closed subcategories is image-closed and is
therefore the meet, while \(\icl(\add\bigcup_i\ZZZ_i)\) is the smallest
image-closed subcategory containing every \(\ZZZ_i\) by
Corollary~\ref{cor:image-closure} and is therefore the join.

(2) Since \(C=\ind\ZZZ\) is finite and \(\icl\)-closed, restricting \(\icl\)
to subsets of \(C\) defines a finite convex geometry.  By Edelman and
Jamison \cite[Theorem~2.1(d)--(e)]{EdelmanJamison}, we have
\(C=\icl(\ex(C))\), and every subset \(S\subseteq C\) with \(\icl(S)=C\)
must contain \(\ex(C)\) because no \(x\in\ex(C)\) belongs to
\(\icl(C\setminus\{x\})\).  Thus \(\ex(C)\) is the unique smallest generating
set of \(C\), and \(\ex(C)=\PZ(\ZZZ)\cup\IZ(\ZZZ)\) by
Proposition~\ref{prop:extreme-points}.

(3) Put \(C=\ind\ZZZ'\) and \(D=\ind\ZZZ\).  For every \(x\in D\setminus C\),
we have \(C\subsetneq\icl(C\cup\{x\})\subseteq D\); since
\(\ZZZ'\subsetneq\ZZZ\) is a covering relation in \(\LI(\Lambda)\), it
follows that \(\icl(C\cup\{x\})=D\).  If
\(D\setminus C\) contained two distinct elements \(x\) and \(y\), we would
have \(y\in\icl(C\cup\{x\})\) and \(x\in\icl(C\cup\{y\})\), contradicting the
anti-exchange property at \(C\).  Thus \(|D\setminus C|=1\), and when
\(\ind\Lambda\) is finite, \(\ZZZ\mapsto|\ind\ZZZ|\) is a rank function on
\(\LI(\Lambda)\).
\end{proof}
\section{Images and closure conditions for interval modules}
\label{sec:intervals}

In this section, we characterize image-closed subcategories for
\(\Lambda_n\), with arrows \(1\to2\to\cdots\to n\), in terms of support intervals
(Theorem~\ref{thm:intersection-criterion}), reducing the second count in
Theorem~\ref{thm:intro-enumeration} to an enumeration of interval families.

\begin{definition}\label{def:interval-board}
For \(1\leq a\leq d\leq n\), the \emph{interval module} \(\M{a}{d}\) is
one-dimensional at the vertices \(a,a+1,\ldots,d\), zero at all other
vertices, and has identity maps along the arrows between consecutive vertices
in \([a,d]\).  By convention, we also set \(\M{a}{a-1}=0\) for
\(1\leq a\leq n+1\).
We call the modules with fixed left endpoint \(a\) \emph{row \(a\)}, and
those with fixed right endpoint \(d\) \emph{column \(d\)}.
\end{definition}

The nonzero interval modules \(\M{a}{d}\) (\(1\leq a\leq d\leq n\)) form a
complete set of representatives of the isomorphism classes of indecomposable
\(\Lambda_n\)-modules, and we often identify \(\M{a}{d}\) with its support
interval \([a,d]\).  The modules in row \(a\) and in column \(d\) are
\[
 \begin{aligned}
  \text{row }a&:\quad \M{a}{a},\M{a}{a+1},\ldots,\M{a}{n},\\
  \text{column }d&:\quad \M{1}{d},\M{2}{d},\ldots,\M{d}{d}.
 \end{aligned}
\]

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=13mm,y=13mm,>=Stealth]
\node (m11) at (6,0) {$\sst{1}$};
\node (m12) at (5,1) {$\sst{1\\2}$};
\node (m13) at (4,2) {$\sst{1\\2\\3}$};
\node (m14) at (3,3) {$\sst{1\\2\\3\\4}$};
\node (m22) at (4,0) {$\sst{2}$};
\node (m23) at (3,1) {$\sst{2\\3}$};
\node (m24) at (2,2) {$\sst{2\\3\\4}$};
\node (m33) at (2,0) {$\sst{3}$};
\node (m34) at (1,1) {$\sst{3\\4}$};
\node (m44) at (0,0) {$\sst{4}$};
\draw[dotted,->] (m11)--(m22);
\draw[->] (m12)--(m11);
\draw[->] (m22)--(m12);
\draw[dotted,->] (m12)--(m23);
\draw[->] (m13)--(m12);
\draw[->] (m23)--(m13);
\draw[dotted,->] (m13)--(m24);
\draw[->] (m14)--(m13);
\draw[->] (m24)--(m14);
\draw[dotted,->] (m22)--(m33);
\draw[->] (m23)--(m22);
\draw[->] (m33)--(m23);
\draw[dotted,->] (m23)--(m34);
\draw[->] (m24)--(m23);
\draw[->] (m34)--(m24);
\draw[dotted,->] (m33)--(m44);
\draw[->] (m34)--(m33);
\draw[->] (m44)--(m34);
\end{tikzpicture}
\caption{The Auslander--Reiten quiver of \(\mod\Lambda_4\). A stack
\(\sst{a\\a+1\\\vdots\\d}\) denotes \(\M{a}{d}\), with top \(S_a\)
and socle \(S_d\). Solid arrows are irreducible maps; dotted arrows point
from a nonprojective module \(X\) to \(\tau X\).}
\label{fig:interval-board}
\end{figure}

Rows and columns are the two families of sectional paths in
Figure~\ref{fig:interval-board}: row arrows are the quotients
\(\M{a}{d}\to\M{a}{d-1}\), and column arrows are the inclusions
\(\M{a+1}{d}\to\M{a}{d}\).  Diagrams of a single subcategory show the full
ambient Auslander--Reiten quiver, with black vertices for indecomposable
modules in the subcategory and open vertices for the remaining
indecomposable modules.  The quivers for \(\Lambda_2\) and \(\Lambda_3\) use
the same arrangement, restricted to composition factors \(1,2\) and
\(1,2,3\), respectively.

\subsection{Morphisms and images of interval modules}

We compute morphisms and their images from the endpoints of the source
\(\M{c}{d}\) and target \(\M{a}{b}\).

\begin{proposition}[{\cite[Proposition~6.1]{ThomasTamari}}]\label{prop:interval-hom}
For two indecomposable modules \(\M{c}{d}\) and \(\M{a}{b}\),
\[
 \dim_{\kk}\Hom_{\Lambda_n}(\M{c}{d},\M{a}{b})=
 \begin{cases}
 1,&a\leq c\leq b\leq d,\\
 0,&\text{otherwise}.
 \end{cases}
\]
When \(\Hom_{\Lambda_n}(\M{c}{d},\M{a}{b})\neq 0\), every nonzero morphism
\(\M{c}{d}\to\M{a}{b}\) has image \(\M{c}{b}\), the interval module
supported on \([c,d]\cap[a,b]\).
\end{proposition}

\begin{proof}
At each vertex \(i\in[c,d]\cap[a,b]\), a morphism \(\M{c}{d}\to\M{a}{b}\) acts
by a scalar, and by commutativity along the arrows \(i\to i+1\) on the overlap,
all these scalars are equal.  Commutativity at the arrows entering
\(\max(a,c)\) and leaving \(\min(b,d)\) requires \(a\leq c\) and \(b\leq d\)
for this scalar to be nonzero, in which case the image is supported on
\([c,b]=[c,d]\cap[a,b]\).
\end{proof}

\subsection{Closure conditions on support intervals}

We translate the three closure conditions of
Definition~\ref{def:image-closed} into conditions on support intervals.

A submodule of \(\M{a}{d}\) is spanned by a subset of its
nonzero vertex spaces that is closed under the arrows \(i\to i+1\), and is
therefore supported on a suffix \([\ell,d]\) of \([a,d]\); the corresponding
quotient is supported on the complementary prefix \([a,\ell-1]\).  Consequently,
 the nonzero quotients and submodules of \(\M{a}{d}\) are indecomposable and
form chains:
\begin{equation}\label{eq:interval-quot-sub}
 \begin{array}{ll}
 \text{indecomposable quotients of }\M{a}{d}:
     &\M{a}{\ell}\quad(a\leq \ell\leq d),\\[2mm]
 \text{indecomposable submodules of }\M{a}{d}:
     &\M{\ell}{d}\quad(a\leq \ell\leq d).
 \end{array}
\end{equation}
Thus, if a quotient-closed subcategory contains \(\M{a}{d}\), it contains
\(\M{a}{\ell}\) for every \(a\leq \ell\leq d\); dually, if a submodule-closed
subcategory contains \(\M{a}{d}\), it contains \(\M{\ell}{d}\) for every
\(a\leq \ell\leq d\).  In Figure~\ref{fig:interval-board}, the indecomposable
modules in a quotient-closed subcategory therefore form a prefix \(\M{a}{a},\ldots,\M{a}{b}\) in each row \(a\),
and those in a submodule-closed subcategory form a suffix
\(\M{c}{d},\ldots,\M{d}{d}\) in each column \(d\).

To apply these interval chains to arbitrary direct sums, we use the
following reduction to a single summand, noted by Campanini, Fedele, and
Y{\i}ld{\i}r{\i}m \cite[Section~4.4]{CFY} and in the proof of
\cite[Theorem~4.2]{EnomotoEquidistribution}.

\begin{lemma}\label{lem:summand-reduction}
Let \(Y\) be an indecomposable \(\Lambda_n\)-module and let
\(X=\bigoplus_{i}X_i\) be a finite direct sum of \(\Lambda_n\)-modules.
\begin{enumerate}
\item If \(Y\) is a quotient of \(X\), then \(Y\) is a quotient of some
      \(X_i\).
\item If \(Y\) is a submodule of \(X\), then \(Y\) is a submodule of some
      \(X_i\).
\end{enumerate}
\end{lemma}

\begin{proof}
By~\eqref{eq:interval-quot-sub}, the submodules of \(Y\) form a finite chain
under inclusion and therefore have a largest and a smallest member in any
nonempty collection.

(1) Let \(g\colon X\twoheadrightarrow Y\) be an epimorphism.  The images
\(g(X_i)\) are submodules of \(Y\) whose sum is \(Y\), so the largest of
them equals \(Y\) and the restriction of \(g\) to the corresponding summand
\(X_i\) is an epimorphism.

(2) Let \(h\colon Y\hookrightarrow X\) be a monomorphism with components
\(h_i\colon Y\to X_i\).  The kernels \(\Ker h_i\) are submodules of \(Y\)
whose intersection is \(\Ker h=0\), so the smallest of them is zero and the
corresponding \(h_i\) is a monomorphism.
\end{proof}

We now characterize image-closed subcategories of \(\mod\Lambda_n\) by their
families of support intervals.

\begin{definition}\label{def:interval-family}
For a subcategory \(\CC\subseteq\mod \Lambda_n\), write
\[
 \II(\CC)=\{[a,d]:\M{a}{d}\in\ind\CC\}
\]
for its family of support intervals.
A family \(\DD\) of intervals is \emph{intersection-closed} if
\[
 I,J\in\DD,\quad I\cap J\neq\varnothing
 \quad\Longrightarrow\quad I\cap J\in\DD.
\]
The empty family is intersection-closed.
\end{definition}

\begin{theorem}\label{thm:intersection-criterion}
Let \(\CC\) be a subcategory of \(\mod \Lambda_n\).
Then \(\CC\) is image-closed if and only if
\(\II(\CC)\) is intersection-closed.
\end{theorem}

\begin{proof}
\((\Rightarrow)\): Suppose that \(\CC\) is image-closed, and let
\(I,J\in\II(\CC)\) satisfy \(I\cap J\neq\varnothing\).  If one of \(I\) and
\(J\) contains the other, then \(I\cap J\) is already one of them and
belongs to \(\II(\CC)\).  Otherwise, after interchanging \(I\) and \(J\) if
necessary, we may write
\[
                I=[a,b],\qquad J=[c,d],\qquad
                a<c\leq b<d.
\]
Since \(a\leq c\leq b\leq d\), Proposition~\ref{prop:interval-hom} gives a
nonzero morphism \(\M{c}{d}\to\M{a}{b}\) with image \(\M{c}{b}\).  Because
\(\M{c}{d},\M{a}{b}\in\CC\) and \(\CC\) is image-closed, we obtain
\(\M{c}{b}\in\CC\), so \(I\cap J=[c,b]\in\II(\CC)\).

\((\Leftarrow)\): Assume that \(\II(\CC)\) is intersection-closed, let
\(f\colon X\to Y\) be a morphism with \(X,Y\in\CC\), and let \(\M{a}{b}\) be
an indecomposable direct summand of \(\Image f\).  Composing the canonical
epimorphism \(X\twoheadrightarrow\Image f\) with the projection onto
\(\M{a}{b}\) exhibits \(\M{a}{b}\) as a quotient of \(X\).  Writing
\(X=\bigoplus_i X_i\) with each \(X_i\in\ind\CC\), we see from
Lemma~\ref{lem:summand-reduction}(1) and~\eqref{eq:interval-quot-sub} that
\(\M{a}{b}\) is a quotient of some \(X_i=\M{a}{d}\) with \(d\geq b\), so
\([a,d]\in\II(\CC)\).

Dually, composing the inclusion of \(\M{a}{b}\) into \(\Image f\) with
\(\Image f\hookrightarrow Y\) exhibits \(\M{a}{b}\) as a submodule of \(Y\).
Writing \(Y=\bigoplus_j Y_j\) with each \(Y_j\in\ind\CC\), we see from
Lemma~\ref{lem:summand-reduction}(2) and~\eqref{eq:interval-quot-sub} that
\(\M{a}{b}\) is a submodule of some \(Y_j=\M{c}{b}\) with \(c\leq a\), so
\([c,b]\in\II(\CC)\).

Since \(c\leq a\leq b\leq d\), the intersection
\[
                         [a,d]\cap[c,b]=[a,b]
\]
is nonempty and therefore belongs to \(\II(\CC)\), which means
\(\M{a}{b}\in\CC\).  Since every indecomposable direct summand of
\(\Image f\) belongs to \(\CC\), we conclude that \(\Image f\in\CC\).
\end{proof}

For \(n=2\), Theorem~\ref{thm:intersection-criterion} imposes no restriction
on a subset of \(\ind\Lambda_2\).

\begin{example}\label{ex:A2-image}
The category \(\mod\Lambda_2\) has three indecomposable modules up to
isomorphism: the simple modules \(\M{1}{1}\) and \(\M{2}{2}\), and the
projective-injective module \(\M{1}{2}\).  Every subset of
\(\{[1,1],[1,2],[2,2]\}\) is intersection-closed because the only nonempty
intersections of distinct intervals in \(\{1,2\}\) are inclusions.  Hence
\(\LI(\Lambda_2)\) is the Boolean lattice of subsets of a three-element set
and has \(2^3=8\) elements.
\end{example}

For \(n=3\), Theorem~\ref{thm:intersection-criterion} restricts which
families of intervals can occur, as the next example shows.

\begin{example}\label{ex:A3-obstruction}
In \(\mod\Lambda_3\), the subcategory \(\CC=\add\{\M{1}{2},\M{2}{3}\}\) has
support family \(\II(\CC)=\{[1,2],[2,3]\}\).  Since
\([1,2]\cap[2,3]=[2,2]\notin\II(\CC)\),
Theorem~\ref{thm:intersection-criterion} shows that \(\CC\) is not
image-closed; indeed, the nonzero morphism \(\M{2}{3}\to\M{1}{2}\) has image
\(\M{2}{2}\notin\CC\).  Adjoining \(\M{2}{2}\) produces the image closure
\(\icl\CC=\add\{\M{1}{2},\M{2}{2},\M{2}{3}\}\).
\end{example}

Table~\ref{tab:A3-images} lists all image-closed subcategories of
\(\mod\Lambda_3\), grouped by size.  Following the convention of
Figure~\ref{fig:interval-board}, each diagram has \(\M13\) at the top,
\(\M23\) and \(\M12\) in the middle from left to right, and
\(\M33,\M22,\M11\) along the bottom.  Among the six indecomposable
modules, the only restriction is that including both \(\M12\) and \(\M23\)
requires \(\M22\), while the remaining three modules can be chosen
independently; thus the table has \((8-1)\cdot8=56\) entries.  The numbers
of image-closed subcategories of sizes \(0,1,\ldots,6\) are
\(1,6,14,17,12,5,1\), so their generating polynomial by size is
\[
                         1+6q+14q^2+17q^3+12q^4+5q^5+q^6.
\]
By Theorem~\ref{thm:intro-enumeration}, this equals \(C_4(1,q)\), whose
coefficients are also listed in Table~\ref{tab:coefficients}.
\begin{longtable}{c c c c c c c c}
\caption{All \(56\) image-closed subcategories of \(\mod\Lambda_3\), grouped by size.}\label{tab:A3-images}\\
\toprule Size&\multicolumn{7}{c}{Subcategories}\\\midrule
\endfirsthead
\toprule Size&\multicolumn{7}{c}{Subcategories}\\\midrule
\endhead
\bottomrule\endfoot
\(0\) & \ARmini{3}{} &  &  &  &  &  & \\
\midrule
\(1\) & \ARmini{3}{1/1} & \ARmini{3}{1/2} & \ARmini{3}{1/3} & \ARmini{3}{2/2} & \ARmini{3}{2/3} & \ARmini{3}{3/3} & \\
\midrule
\(2\) & \ARmini{3}{1/1,1/2} & \ARmini{3}{1/1,1/3} & \ARmini{3}{1/2,1/3} & \ARmini{3}{1/1,2/2} & \ARmini{3}{1/2,2/2} & \ARmini{3}{1/3,2/2} & \ARmini{3}{1/1,2/3}\\
 & \ARmini{3}{1/3,2/3} & \ARmini{3}{2/2,2/3} & \ARmini{3}{1/1,3/3} & \ARmini{3}{1/2,3/3} & \ARmini{3}{1/3,3/3} & \ARmini{3}{2/2,3/3} & \ARmini{3}{2/3,3/3}\\
\midrule
\(3\) & \ARmini{3}{1/1,1/2,1/3} & \ARmini{3}{1/1,1/2,2/2} & \ARmini{3}{1/1,1/3,2/2} & \ARmini{3}{1/2,1/3,2/2} & \ARmini{3}{1/1,1/3,2/3} & \ARmini{3}{1/1,2/2,2/3} & \ARmini{3}{1/2,2/2,2/3}\\
 & \ARmini{3}{1/3,2/2,2/3} & \ARmini{3}{1/1,1/2,3/3} & \ARmini{3}{1/1,1/3,3/3} & \ARmini{3}{1/2,1/3,3/3} & \ARmini{3}{1/1,2/2,3/3} & \ARmini{3}{1/2,2/2,3/3} & \ARmini{3}{1/3,2/2,3/3}\\
 & \ARmini{3}{1/1,2/3,3/3} & \ARmini{3}{1/3,2/3,3/3} & \ARmini{3}{2/2,2/3,3/3} &  &  &  & \\
\midrule
\(4\) & \ARmini{3}{1/1,1/2,1/3,2/2} & \ARmini{3}{1/1,1/2,2/2,2/3} & \ARmini{3}{1/1,1/3,2/2,2/3} & \ARmini{3}{1/2,1/3,2/2,2/3} & \ARmini{3}{1/1,1/2,1/3,3/3} & \ARmini{3}{1/1,1/2,2/2,3/3} & \ARmini{3}{1/1,1/3,2/2,3/3}\\
 & \ARmini{3}{1/2,1/3,2/2,3/3} & \ARmini{3}{1/1,1/3,2/3,3/3} & \ARmini{3}{1/1,2/2,2/3,3/3} & \ARmini{3}{1/2,2/2,2/3,3/3} & \ARmini{3}{1/3,2/2,2/3,3/3} &  & \\
\midrule
\(5\) & \ARmini{3}{1/1,1/2,1/3,2/2,2/3} & \ARmini{3}{1/1,1/2,1/3,2/2,3/3} & \ARmini{3}{1/1,1/2,2/2,2/3,3/3} & \ARmini{3}{1/1,1/3,2/2,2/3,3/3} & \ARmini{3}{1/2,1/3,2/2,2/3,3/3} &  & \\
\midrule
\(6\) & \ARmini{3}{1/1,1/2,1/3,2/2,2/3,3/3} &  &  &  &  &  & \\
\end{longtable}
\section{Classification of pretorsion theories by cutoff coordinates}
\label{sec:pretorsion-type-a}

Having characterized the image-closed subcategories of \(\mod\Lambda_n\) by
their support intervals, we now classify the pretorsion theories that
realize them.  We parametrize the quotient-closed and submodule-closed
subcategories of \(\mod\Lambda_n\) by cutoff coordinates on the rows and
columns of Figure~\ref{fig:interval-board}—where row \(a\) fixes the left
endpoint of \(\M{a}{d}\) and column \(d\) fixes its right endpoint—and
characterize which pairs form pretorsion theories
(Theorem~\ref{thm:staircase-pair}).

\subsection{Row and column cutoffs for quotient and submodule closure}

The parametrization in \cite[Theorem~4.2(1)--(2)]{EnomotoEquidistribution}
specializes here to choosing an endpoint on each row or column of
Figure~\ref{fig:interval-board}, as follows.

\begin{proposition}
\label{prop:staircase}
There are bijections
\begin{align*}
 \prod_{a=1}^n\{a-1,a,\ldots,n\}&\longrightarrow\LQ(\Lambda_n),
 &\beta&\longmapsto\Tvec{\beta},\\
 \prod_{d=1}^n\{1,2,\ldots,d+1\}&\longrightarrow\LS(\Lambda_n),
 &\alpha&\longmapsto\Fvec{\alpha},
\end{align*}
where
\begin{align}
 \M{a}{d}\in\Tvec{\beta}&\quad\Longleftrightarrow\quad d\leq\beta_a,
 \label{eq:T-beta}\\
 \M{a}{d}\in\Fvec{\alpha}&\quad\Longleftrightarrow\quad
                                      \alpha_d\leq a.
 \label{eq:F-alpha}
\end{align}
The values \(\beta_a=a-1\) and \(\alpha_d=d+1\) specify an empty row
and an empty column, respectively.  Moreover,
\[
 \Tvec{\beta}\subseteq\Tvec{\beta'}\Longleftrightarrow\beta\leq\beta',
 \qquad
 \Fvec{\alpha}\supseteq\Fvec{\alpha'}
                         \Longleftrightarrow\alpha\leq\alpha'.
\]
\end{proposition}

\begin{proof}
By~\eqref{eq:interval-quot-sub}, a set of indecomposable modules is closed
under nonzero quotients if and only if its intersection with each row is an
initial segment \([a,a],\ldots,[a,\beta_a]\), and closed under nonzero
submodules if and only if its intersection with each column is a final
segment \([\alpha_d,d],\ldots,[d,d]\).  Lemma~\ref{lem:summand-reduction} extends
both characterizations to finite direct sums, giving the two bijections, and
the inclusion relations follow directly from these segments.
\end{proof}

\begin{definition}\label{def:row-column-cutoff}
For \(\Tvec{\beta}\) in Proposition~\ref{prop:staircase}, the coordinate
\(\beta_a\) is the \emph{row cutoff} of row \(a\): it is the right endpoint
of the last interval of row \(a\) belonging to \(\Tvec{\beta}\), with
\(\beta_a=a-1\) when row \(a\) contains no object of \(\Tvec{\beta}\).  For
\(\Fvec{\alpha}\), the coordinate \(\alpha_d\) is the \emph{column cutoff}
of column \(d\): the intervals in column \(d\) belonging to \(\Fvec{\alpha}\)
have left endpoints \(\alpha_d,\alpha_d+1,\ldots,d\), with \(\alpha_d=d+1\)
when column \(d\) contains no object of \(\Fvec{\alpha}\).  We call
\((\beta,\alpha)\) the pair of \emph{cutoff coordinates} of
\((\Tvec{\beta},\Fvec{\alpha})\).  When \(\alpha_d>1\), we call
\(e=\alpha_d-1\) the \emph{pivot row} in column \(d\): it is the last row
excluded from \(\Fvec{\alpha}\) in that column, and
Theorem~\ref{thm:staircase-pair} compares it with the row cutoffs of
\(\Tvec{\beta}\).
\end{definition}

By Proposition~\ref{prop:staircase}, the order~\eqref{eq:PT-order} on
\(\PT(\Lambda_n)\) corresponds to the coordinatewise order on cutoff
coordinates \((\beta,\alpha)\).  However, the coordinatewise minimum and
maximum of two cutoff pairs in \(\PT(\Lambda_n)\) need not define pretorsion
theories, so the meet and join of Theorem~\ref{thm:PT-lattice} are not given
by coordinatewise minimum and maximum in general (see
Example~\ref{ex:coordinate-operations}).

Next, we express the trace and the reject of an interval module in terms of
cutoff coordinates.  For a vector \(\beta\) and \(1\leq d\leq n\), put
\begin{equation}\label{eq:E-d}
                   E_d(\beta)=\{a\leq d:\beta_a\geq d\}.
\end{equation}
By~\eqref{eq:T-beta}, \(E_d(\beta)\) is the set of row indices \(a\leq d\)
such that \(\M{a}{d}\in\Tvec{\beta}\).

\begin{lemma}\label{lem:interval-trace-reject}
Let \(X=\M{c}{d}\).  For \(\TT=\Tvec{\beta}\),
\[
 t_{\TT}(X)=
 \begin{cases}
  \M{u}{d},&u=\min(E_d(\beta)\cap[c,d]),\\
  0,&E_d(\beta)\cap[c,d]=\varnothing.
 \end{cases}
\]
For \(\FF=\Fvec{\alpha}\),
\[
 r_{\FF}(X)=\M{b+1}{d},\qquad
 b=\max\bigl(\{j\in[c,d]:\alpha_j\leq c\}\cup\{c-1\}\bigr).
\]
Here \(b=d\) gives \(\M{d+1}{d}=0\) by Definition~\ref{def:interval-board},
while \(b=c-1\) gives \(r_{\FF}(X)=X\) when no nonzero quotient of \(X\)
belongs to \(\FF\).
\end{lemma}

\begin{proof}
By Proposition~\ref{prop:trace-reject-properties}(1), \(t_{\TT}(X)\) is the
largest submodule of \(X\) belonging to \(\TT\).  By~\eqref{eq:interval-quot-sub},
the nonzero submodules of \(X=\M{c}{d}\) are the modules \(\M{a}{d}\) with
\(c\leq a\leq d\), and \(\M{a}{d}\in\TT\) if and only if \(a\in E_d(\beta)\).
Among these submodules, the largest has the smallest left endpoint \(u\),
giving the formula for \(t_{\TT}(X)\).

Dually, by Proposition~\ref{prop:trace-reject-properties}(2),
\(X/r_{\FF}(X)\) is the largest quotient of \(X\) belonging to \(\FF\).  The
nonzero quotients of \(X\) are the modules \(\M{c}{j}\) with
\(c\leq j\leq d\), and \(\M{c}{j}\in\FF\) if and only if \(\alpha_j\leq c\).
The largest such quotient has the greatest right endpoint \(b\) and kernel
\(r_{\FF}(X)=\M{b+1}{d}\); when no such quotient exists, \(X/r_{\FF}(X)=0\)
and \(r_{\FF}(X)=X\), which corresponds to \(b=c-1\).
\end{proof}

\subsection{Cutoff pairs defining pretorsion theories}

We characterize the cutoff pairs of pretorsion theories one column at a
time, using the earliest row whose cutoff is \(d-1\).

For \(1\leq d\leq n\), define
\begin{equation}\label{eq:s-d}
 s_d(\beta)=\min\bigl(\{a:\beta_a=d-1\}\cup\{d+1\}\bigr).
\end{equation}
Since \(\beta_a\geq a-1\geq d\) for all \(a>d\), any row \(a\) with
\(\beta_a=d-1\) satisfies \(a\leq d\).  For \(a<d\), the equality
\(\beta_a=d-1\) means that \(\M{a}{d-1}\) is the last module of row \(a\) in
\(\Tvec{\beta}\); for \(a=d\), it means that row \(d\) is empty in
\(\Tvec{\beta}\).  When no row has cutoff \(d-1\), we have \(s_d(\beta)=d+1\).
If \(\beta_a\geq d\), then \(\M{a}{d}\) belongs to \(\Tvec{\beta}\), whereas
if \(\beta_a<d-1\), row \(a\) already ends before column \(d-1\).  The new
obstruction in column \(d\) therefore comes from rows with cutoff
\(\beta_a=d-1\): for such a row, \(\M{a}{d}\notin\Tvec{\beta}\) while its
quotient \(\M{a}{d-1}\) belongs to \(\Tvec{\beta}\), which can cause the
equality \(t(X/r(X))=t(X)/r(X)\) to fail for \(X=\M{a}{d}\).

The following example exhibits this failure for a row preceding the pivot.

\begin{example}\label{ex:earlier-row-failure}
In \(\mod\Lambda_2\), take \(\beta=(1,2)\) and \(\alpha=(1,3)\), so that
\(\TT=\Tvec{\beta}=\add\{\M11,\M22\}\) and
\(\FF=\Fvec{\alpha}=\add\M11\).  In column \(2\), the pivot row is
\(e=\alpha_2-1=2\in E_2(\beta)\), but the earlier row \(a=1\) has cutoff
\(\beta_1=1=2-1\).  Writing
\(t=t_{\TT}\) and \(r=r_{\FF}\), we find from
Lemma~\ref{lem:interval-trace-reject} that \(X=\M12\) satisfies
\[
 t(X)=r(X)=r(t(X))=\M22,\qquad
 t(X/r(X))=t(\M11)=\M11\neq0=t(X)/r(X),
\]
so the second equality in~\eqref{eq:trace-reject} fails.
\end{example}

For a pair \((\Tvec{\beta},\Fvec{\alpha})\), if \(\alpha_d=e+1>1\), then
every \(\M{a}{d}\) with \(a>e\) belongs to \(\Fvec{\alpha}\), so both
equalities in~\eqref{eq:trace-reject} hold automatically on these modules.
The criterion in column \(d\) therefore concerns only the pivot row \(e\)
and the rows preceding it.

\begin{theorem}\label{thm:staircase-pair}
The pair \((\Tvec{\beta},\Fvec{\alpha})\) is a pretorsion theory if and only
if, for every \(1\leq d\leq n\), either \(\alpha_d=1\), or the row
\(e=\alpha_d-1\) has cutoff at least \(d\) while no earlier row has cutoff
exactly \(d-1\):
\begin{equation}\label{eq:pair-criterion}
 \beta_e\geq d,
 \qquad
 \beta_a\neq d-1\quad(1\leq a<e).
\end{equation}
Equivalently,
\begin{equation}\label{eq:allowed-alpha}
 \alpha_d\in
 \{1\}\cup\{e+1:e\in E_d(\beta),\ e<s_d(\beta)\}.
\end{equation}
\end{theorem}

\begin{remark}\label{rem:fixed-beta-chains}
For a fixed \(\beta\), each allowed set in~\eqref{eq:allowed-alpha} depends
only on \(\beta\) and \(d\), so the second components \(\Fvec{\alpha}\) for
which \((\Tvec{\beta},\Fvec{\alpha})\) is a pretorsion theory, ordered by
reverse inclusion, form the product of these chains of allowed values; see
Example~\ref{ex:A2-partners}.
\end{remark}

Before proving Theorem~\ref{thm:staircase-pair}, we illustrate the
condition~\eqref{eq:pair-criterion} for \(n=2\) and \(n=3\).

\begin{example}\label{ex:A2-partners}
In \(\mod\Lambda_2\), fix
\[
 \beta=(2,1),\qquad
 \Tvec{\beta}=\add\{\M11,\M12\},
\]
so that row \(1\) of \(\Tvec{\beta}\) is full (\(\beta_1=2\)) and row \(2\)
is empty (\(\beta_2=1\)).  In column \(1\), we have \(E_1(\beta)=\{1\}\) and
\(s_1(\beta)=2\) since no row has \(\beta_a=0\); in column \(2\), we have
\(E_2(\beta)=\{1\}\) and \(s_2(\beta)=2\) since \(\beta_2=1\).  Because
\(1<s_d(\beta)\) for \(d=1,2\), condition~\eqref{eq:allowed-alpha}
permits \(\alpha_d\in\{1,2\}\) in both columns, yielding four pretorsion
theories with first component \(\Tvec{\beta}\):
\begin{center}
\begin{tabular}{c|c|c}
\toprule
\(\alpha\)&\(\Fvec{\alpha}\)&\(\Tvec{\beta}\cap\Fvec{\alpha}\)\\
\midrule
\((1,1)\)&\(\mod \Lambda_2\)&\(\add\{\M11,\M12\}\)\\
\((2,1)\)&\(\add\{\M12,\M22\}\)&\(\add\M12\)\\
\((1,2)\)&\(\add\{\M11,\M22\}\)&\(\add\M11\)\\
\((2,2)\)&\(\add\M22\)&\(0\)\\
\bottomrule
\end{tabular}
\end{center}
The product of the two chains \(\{1,2\}\) in
Remark~\ref{rem:fixed-beta-chains} gives a Boolean square in
\(\PT(\Lambda_2)\):
\[
\begin{tikzpicture}[baseline=-2mm,node distance=6mm and 12mm]
 \node (d) {\((2,2)\)};
 \node (b) [below left=of d] {\((2,1)\)};
 \node (c) [below right=of d] {\((1,2)\)};
 \node (a) [below right=of b] {\((1,1)\)};
 \draw (d)--(b)--(a)--(c)--(d);
\end{tikzpicture}
\]
Campanini and Fedele \cite[Remark~6.3]{CampaniniFedele} use the pair
with \(\alpha=(1,2)\) to exhibit a pretorsion theory whose second
component, \(\add\{\M11,\M22\}\), is not closed under extensions.
\end{example}

\begin{example}\label{ex:A3-cutoffs}
Figure~\ref{fig:alpha-beta-A3} displays the pretorsion theory in
\(\mod\Lambda_3\) with cutoff coordinates \(\beta=(3,2,2)\) and
\(\alpha=(2,1,2)\).  Column \(2\) has \(\alpha_2=1\), while columns \(1\)
and \(3\) have pivot row \(e=\alpha_d-1=1\) with \(\beta_1=3\geq d\) and no
earlier rows, so~\eqref{eq:pair-criterion} holds in every column.
\end{example}
\begin{figure}[ht]
\centering
\begin{tabular}{ccc}
 \(\TT\)&\(\FF\)&\(\TT\cap\FF\)\\
 \ARmini{3}{1/1,1/2,1/3,2/2}&
 \ARmini{3}{1/2,2/2,2/3,3/3}&
 \ARmini{3}{1/2,2/2}
\end{tabular}
\caption{The pretorsion theory with \(\beta=(3,2,2)\) and
\(\alpha=(2,1,2)\); black vertices mark membership in each subcategory,
and open vertices mark the remaining indecomposables.}
\label{fig:alpha-beta-A3}
\end{figure}

\begin{proof}[Proof of Theorem~\ref{thm:staircase-pair}]
Put \(\TT=\Tvec{\beta}\), \(\FF=\Fvec{\alpha}\), \(t=t_{\TT}\), and
\(r=r_{\FF}\).  By Theorem~\ref{thm:trace-reject}, \((\TT,\FF)\) is a
pretorsion theory if and only if \(r(X)=r(t(X))\) and
\(t(X/r(X))=t(X)/r(X)\) for every module \(X\).

\((\Rightarrow)\): Suppose that \((\TT,\FF)\) is a pretorsion theory, and
fix a column \(d\) with \(e=\alpha_d-1>0\).  For \(X=\M{e}{d}\), we have
\(X\notin\FF\) by~\eqref{eq:F-alpha}, whereas every nonzero proper submodule
\(\M{c}{d}\) (\(e+1\leq c\leq d\)) belongs to \(\FF\).  If
\(t(X)\subsetneq X\), then \(t(X)\in\FF\), so \(r(t(X))=0\neq r(X)\) by
Proposition~\ref{prop:trace-reject-properties}(3), contrary
to~\eqref{eq:trace-reject}.  Hence \(t(X)=X\), which gives \(\beta_e\geq d\).

Suppose next that \(1\leq a<e\) and \(\beta_a=d-1\).  For \(X=\M{a}{d}\), we
have \(X\notin\TT\) since \(\beta_a=d-1<d\), and \(X\notin\FF\) since
\(a<e<\alpha_d\).  By Proposition~\ref{prop:trace-reject-properties}(3) and
the equality \(r(X)=r(t(X))\) in~\eqref{eq:trace-reject}, it follows that
\[
                    0\neq r(X)\subseteq t(X)\subsetneq X.
\]
Thus \(X/r(X)=\M{a}{b}\) for some \(a\leq b\leq d-1=\beta_a\), so
\(X/r(X)\in\TT\).  Consequently,
\[
             t(X/r(X))=X/r(X)\neq t(X)/r(X),
\]
again contradicting~\eqref{eq:trace-reject}.  This proves
condition~\eqref{eq:pair-criterion}.

\((\Leftarrow)\): Suppose that condition~\eqref{eq:pair-criterion} holds in
every column \(d\) with \(\alpha_d>1\).  Since \(t\) and \(r\) commute with
finite direct sums by Proposition~\ref{prop:trace-reject-properties}(4), it
suffices to verify~\eqref{eq:trace-reject} for each indecomposable module
\(X=\M{c}{d}\).  If \(X\in\FF\), then \(t(X)\in\FF\) by submodule closure,
so \(r(X)=r(t(X))=0\) and both equalities in~\eqref{eq:trace-reject} hold.

Assume now that \(X=\M{c}{d}\notin\FF\), so that \(\alpha_d>c\).
The pivot row \(e=\alpha_d-1\) in column \(d\) satisfies \(c\leq e\leq d\) and
\(\beta_e\geq d\), so \(e\in E_d(\beta)\cap[c,d]\).  We may therefore put
\[
 u=\min(E_d(\beta)\cap[c,d]),\qquad
 b=\max\bigl(\{\ell\in[c,d]:\alpha_\ell\leq c\}\cup\{c-1\}\bigr),
\]
and Lemma~\ref{lem:interval-trace-reject} gives
\[
                 t(X)=\M{u}{d},\qquad r(X)=\M{b+1}{d},
\]
with \(b<d\) because \(\alpha_d>c\).  We first show that
\(\beta_i<b\) for \(c\leq i<u\), ruling out a trace contribution starting
before \(u\) in the quotient \(X/r(X)=\M{c}{b}\).  We then use this bound
to verify the reject equality and the trace equality.

\medskip
\noindent\textbf{Step 1: Excluding earlier rows.}
If a row before \(u\) reached column \(b\), it would end in some column
\(j-1<d\), and condition~\eqref{eq:pair-criterion} in column \(j\) would
produce an even earlier row with a larger cutoff.  We use this observation
to show that
\begin{equation}\label{eq:cutoff-earlier-row}
                         \beta_i<b\qquad(c\leq i<u).
\end{equation}
Suppose for contradiction that~\eqref{eq:cutoff-earlier-row} fails, choose
\(i\in[c,u-1]\) with \(\beta_i\) maximal, so that \(\beta_i\geq b\), and put
\(j=\beta_i+1\).  By the minimality of \(u\), we have \(i\notin E_d(\beta)\)
and hence \(\beta_i<d\), so \(b<j\leq d\); since \(b\geq c-1\), we also have
\(j\geq c\).  The definition of \(b\) then gives \(\alpha_j>c\), so the
pivot row \(e'=\alpha_j-1\) in column \(j\) satisfies \(e'\geq c\) and
\(\beta_{e'}\geq j\).  Since \(\beta_i=j-1\) and \(\beta_{e'}\geq j>\beta_i\),
condition~\eqref{eq:pair-criterion} in column \(j\) implies \(e'<i\).
Thus \(e'\in[c,u-1]\) and \(\beta_{e'}\geq j>\beta_i\), contradicting the
maximality of \(\beta_i\) and proving~\eqref{eq:cutoff-earlier-row}.

\medskip
\noindent\textbf{Step 2: The equality \(r(t(X))=r(X)\).}
From~\eqref{eq:cutoff-earlier-row}, we obtain \(u\leq b+1\): this is
immediate if \(u=c\), while if \(u>c\), the bound
\(\beta_a\geq a-1\) at \(a=u-1\) gives \(u-2\leq\beta_{u-1}<b\).
Hence \(r(X)=\M{b+1}{d}\subseteq\M{u}{d}=t(X)\).  To compute \(r(t(X))\),
note that for \(b<j\leq d\) we have \(\alpha_j>c\); if we also had
\(\alpha_j\leq u\), the pivot row \(e'=\alpha_j-1\) in column \(j\) would
satisfy \(c\leq e'<u\) and \(\beta_{e'}\geq j>b\), contradicting
\eqref{eq:cutoff-earlier-row}.  Thus \(\alpha_j>u\) for all \(b<j\leq d\),
so no quotient \(\M{u}{j}\) of \(t(X)=\M{u}{d}\) with \(j>b\) belongs to
\(\FF\).  If \(b\geq u\), then \(\alpha_b\leq c\leq u\), so
\(\M{u}{b}\in\FF\); if \(b=u-1\), no nonzero quotient of \(\M{u}{d}\)
belongs to \(\FF\).  In both cases, Lemma~\ref{lem:interval-trace-reject}
gives
\[
                         r(t(X))=\M{b+1}{d}=r(X).
\]

\medskip
\noindent\textbf{Step 3: The equality \(t(X/r(X))=t(X)/r(X)\).}
We have
\(X/r(X)=\M{c}{b}\) (which is zero when \(b=c-1\)).  If \(b\geq u\),
then~\eqref{eq:cutoff-earlier-row} shows that \(i\notin E_b(\beta)\) for
\(c\leq i<u\), whereas \(\beta_u\geq d\geq b\) gives \(u\in E_b(\beta)\), so
\(t(\M{c}{b})=\M{u}{b}\) by Lemma~\ref{lem:interval-trace-reject}.  If
\(b=u-1\), then~\eqref{eq:cutoff-earlier-row} gives \(\beta_i<b\) for every
\(c\leq i\leq b\), so \(E_b(\beta)\cap[c,b]=\varnothing\) and
\(t(\M{c}{b})=0\).  With the convention \(\M{u}{u-1}=0\) of
Definition~\ref{def:interval-board}, both cases give
\[
               t(X/r(X))=t(\M{c}{b})=\M{u}{b}=t(X)/r(X).
\]
This establishes~\eqref{eq:trace-reject}, so \((\TT,\FF)\) is a pretorsion
theory.

Finally, for \(\alpha_d=e+1>1\), the inequality \(\beta_e\geq d\) means that
\(e\in E_d(\beta)\).  Since \(s_d(\beta)\notin E_d(\beta)\), the absence of
an earlier row \(a<e\) with \(\beta_a=d-1\) is equivalent to
\(e<s_d(\beta)\).  This proves~\eqref{eq:allowed-alpha}.
\end{proof}

\subsection{All pretorsion theories for two vertices}

The complete list for \(\Lambda_2\) displays both the realizations of each
image-closed subcategory and the order on pretorsion theories.
Table~\ref{tab:A2-complete} groups all \(17\) pretorsion theories by their
intersection \(\ZZZ\), marking the least and greatest realization from
Corollaries~\ref{cor:greatest-realization} and~\ref{cor:least-realization}.
The eight groups correspond to the eight subcategories in
Example~\ref{ex:A2-image}, and Figure~\ref{fig:PT-A2} gives the Hasse diagram.
In each diagram the vertices, from left to right, are
\(\M22,\M12,\M11\), with \(\M12\) at the apex, as in
Figure~\ref{fig:interval-board}; the membership colours are explained in
Table~\ref{tab:A2-complete}.

\begin{table}[htp]
\centering
\caption{All pretorsion theories of \(\Lambda_2\), grouped by their intersection. Blue: \(\TT\) only; red: \(\FF\) only; blue top half and red bottom half: \(\TT\cap\FF\); open: neither. L and G mark the least and greatest realization of each intersection.}\label{tab:A2-complete}
\begin{minipage}[t]{.48\textwidth}\centering
\begin{tabular}[t]{@{}c c c c l@{}}
\toprule No.&\((\TT,\FF)\)&\(\beta\)&\(\alpha\)&\\\midrule
1 & \PTmini{2}{}{1/1,1/2,2/2}{} & \((0,1)\) & \((1,1)\) & L \\
2 & \PTmini{2}{2/2}{1/1}{} & \((0,2)\) & \((1,3)\) &  \\
3 & \PTmini{2}{1/1}{1/2,2/2}{} & \((1,1)\) & \((2,1)\) &  \\
4 & \PTmini{2}{1/1,1/2}{2/2}{} & \((2,1)\) & \((2,2)\) &  \\
5 & \PTmini{2}{1/1,1/2,2/2}{}{} & \((2,2)\) & \((2,3)\) & G \\
\midrule
6 & \PTmini{2}{}{1/2,2/2}{1/1} & \((1,1)\) & \((1,1)\) & L \\
7 & \PTmini{2}{1/2}{2/2}{1/1} & \((2,1)\) & \((1,2)\) &  \\
8 & \PTmini{2}{1/2,2/2}{}{1/1} & \((2,2)\) & \((1,3)\) & G \\
\midrule
9 & \PTmini{2}{1/1}{2/2}{1/2} & \((2,1)\) & \((2,1)\) & L, G \\
\bottomrule\end{tabular}
\end{minipage}%
\hfill
\begin{minipage}[t]{.48\textwidth}\centering
\begin{tabular}[t]{@{}c c c c l@{}}
\toprule No.&\((\TT,\FF)\)&\(\beta\)&\(\alpha\)&\\\midrule
10 & \PTmini{2}{}{1/1,1/2}{2/2} & \((0,2)\) & \((1,1)\) & L \\
11 & \PTmini{2}{1/1}{1/2}{2/2} & \((1,2)\) & \((2,1)\) &  \\
12 & \PTmini{2}{1/1,1/2}{}{2/2} & \((2,2)\) & \((2,2)\) & G \\
\midrule
13 & \PTmini{2}{}{2/2}{1/1,1/2} & \((2,1)\) & \((1,1)\) & L, G \\
\midrule
14 & \PTmini{2}{}{1/2}{1/1,2/2} & \((1,2)\) & \((1,1)\) & L \\
15 & \PTmini{2}{1/2}{}{1/1,2/2} & \((2,2)\) & \((1,2)\) & G \\
\midrule
16 & \PTmini{2}{1/1}{}{1/2,2/2} & \((2,2)\) & \((2,1)\) & L, G \\
\midrule
17 & \PTmini{2}{}{}{1/1,1/2,2/2} & \((2,2)\) & \((1,1)\) & L, G \\
\bottomrule\end{tabular}
\end{minipage}%
\end{table}
\begin{figure}[htp]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,fill=white,rounded corners,inner sep=4pt] (p0) at (0.00,0.00) {\small 1\;\PTmini{2}{}{1/1,1/2,2/2}{}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p1) at (-5.50,6.24) {\small 2\;\PTmini{2}{2/2}{1/1}{}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p2) at (3.50,4.16) {\small 3\;\PTmini{2}{1/1}{1/2,2/2}{}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p3) at (0.00,8.32) {\small 4\;\PTmini{2}{1/1,1/2}{2/2}{}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p4) at (0.00,12.48) {\small 5\;\PTmini{2}{1/1,1/2,2/2}{}{}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p5) at (2.00,2.08) {\small 6\;\PTmini{2}{}{1/2,2/2}{1/1}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p6) at (-2.40,6.24) {\small 7\;\PTmini{2}{1/2}{2/2}{1/1}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p7) at (-3.00,10.40) {\small 8\;\PTmini{2}{1/2,2/2}{}{1/1}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p8) at (2.40,6.24) {\small 9\;\PTmini{2}{1/1}{2/2}{1/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p9) at (-3.00,2.08) {\small 10\;\PTmini{2}{}{1/1,1/2}{2/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p10) at (4.80,6.24) {\small 11\;\PTmini{2}{1/1}{1/2}{2/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p11) at (2.00,10.40) {\small 12\;\PTmini{2}{1/1,1/2}{}{2/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p12) at (-2.40,4.16) {\small 13\;\PTmini{2}{}{2/2}{1/1,1/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p13) at (1.00,4.16) {\small 14\;\PTmini{2}{}{1/2}{1/1,2/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p14) at (-2.40,8.32) {\small 15\;\PTmini{2}{1/2}{}{1/1,2/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p15) at (3.00,8.32) {\small 16\;\PTmini{2}{1/1}{}{1/2,2/2}};
\node[draw,fill=white,rounded corners,inner sep=4pt] (p16) at (0.00,6.24) {\small 17\;\PTmini{2}{}{}{1/1,1/2,2/2}};
\begin{scope}[on background layer]
\draw (p0)--(p5);
\draw (p0)--(p9);
\draw (p1)--(p7);
\draw (p2)--(p8);
\draw (p2)--(p10);
\draw (p3)--(p11);
\draw (p5)--(p2);
\draw (p5)--(p12);
\draw (p5)--(p13);
\draw (p6)--(p3);
\draw (p6)--(p14);
\draw (p7)--(p4);
\draw (p8)--(p3);
\draw (p8)--(p15);
\draw (p9)--(p1);
\draw (p9)--(p13);
\draw (p10)--(p15);
\draw (p11)--(p4);
\draw (p12)--(p6);
\draw (p12)--(p8);
\draw (p12)--(p16);
\draw (p13)--(p10);
\draw (p13)--(p16);
\draw (p14)--(p7);
\draw (p14)--(p11);
\draw (p15)--(p11);
\draw (p16)--(p14);
\draw (p16)--(p15);
\end{scope}
\end{tikzpicture}
\caption{The Hasse diagram of \(\PT(\Lambda_2)\), increasing upwards; numbers and membership colours are as in Table~\ref{tab:A2-complete}.}
\label{fig:PT-A2}
\end{figure}
\FloatBarrier

The following example computes a meet and a join in Figure~\ref{fig:PT-A2}.

\begin{example}\label{ex:coordinate-operations}
Let \(P_i\) denote entry No.~\(i\) of Table~\ref{tab:A2-complete}.
The coordinatewise minimum of
\[
 P_2:\ ((0,2),(1,3)),\qquad P_4:\ ((2,1),(2,2))
\]
is \(((0,1),(1,2))\), which violates~\eqref{eq:pair-criterion} in
column \(2\), since its pivot \(e=1\) has \(\beta_1=0<2\).
Their meet is instead \(P_1\), as seen in Figure~\ref{fig:PT-A2}.
Similarly, the coordinatewise maximum of \(P_2\) and
\(P_3:\ ((1,1),(2,1))\) is \(((1,2),(2,3))\), which
violates~\eqref{eq:pair-criterion} in column \(2\) because its pivot row
\(e=2\) is preceded by row \(1\) with \(\beta_1=1=2-1\).
Their join is \(P_5\).
\end{example}

\FloatBarrier
\section{Enumeration of pretorsion theories by adjoining vertices}
\label{sec:first-gandhi}

In this section, we use the cutoff classification of
Theorem~\ref{thm:staircase-pair} to prove the first identity of
Theorem~\ref{thm:intro-enumeration} (Theorem~\ref{thm:pretorsion-gandhi})
by adjoining one vertex at a time.

\begin{definition}\label{def:pretorsion-statistics}
For \(m\geq1\) and any subcategory \(\CC\subseteq\mod\Lambda_m\), let
\begin{equation}\label{eq:pret-b}
 \rho_m(\CC)=\lvert\{a\in\{1,\ldots,m\}:\M{a}{m}\in\CC\}\rvert
\end{equation}
be the number of indecomposable modules in \(\CC\) whose support ends at
vertex \(m\), and set \(\rho_0(0)=0\).  To each pretorsion theory
\((\TT,\FF)\in\PT(\Lambda_m)\), we assign the weight
\[
 x^{\rho_m(\TT)}q^{|\ind(\TT\cap\FF)|}.
\]
\end{definition}

For \(\TT=\Tvec{\beta}\subseteq\mod\Lambda_m\), the bound \(\beta_a\leq m\)
gives
\(\rho_m(\Tvec{\beta})=\lvert E_m(\beta)\rvert=\lvert\{a\in\{1,\ldots,m\}:\beta_a=m\}\rvert\).
In a fixed column \(d\), the membership rule~\eqref{eq:F-alpha} and the
pivot row of Definition~\ref{def:row-column-cutoff} determine how many
indecomposable modules in column \(d\) belong to
\(\Tvec{\beta}\cap\Fvec{\alpha}\).  If \(\alpha_d=1\), then every
\(\M{a}{d}\) with \(a\in E_d(\beta)\) belongs to
\(\Tvec{\beta}\cap\Fvec{\alpha}\); if \(\alpha_d=e+1>1\), then precisely the
modules \(\M{a}{d}\) with \(a\in E_d(\beta)\) and \(a>e\) belong to
\(\Tvec{\beta}\cap\Fvec{\alpha}\).  Thus the number of indecomposable
modules in column \(d\) of \(\Tvec{\beta}\cap\Fvec{\alpha}\) is
\begin{equation}\label{eq:column-middle-size}
 \begin{cases}
 \lvert E_d(\beta)\rvert,&\alpha_d=1,\\
 \lvert\{a\in E_d(\beta):a>e\}\rvert,&\alpha_d=e+1.
 \end{cases}
\end{equation}

For \(n=1\), the three pretorsion theories of Example~\ref{ex:A1-pretorsion}
have cutoff coordinates
\[
 (\beta,\alpha)=(0,1),\qquad(1,1),\qquad(1,2).
\]
In the first pair, \(E_1(\beta)=\varnothing\), so \(\rho_1(\TT)=0\) and
\(\TT\cap\FF=0\); in the second, \(\rho_1(\TT)=1\) and
\(\lvert\ind(\TT\cap\FF)\rvert=1\); in the third, \(\rho_1(\TT)=1\) and
\(\TT\cap\FF=0\).  Their weights are \(1\), \(xq\), and \(x\), respectively,
and, by~\eqref{eq:intro-B}, their sum is
\begin{equation}\label{eq:A1-pretorsion-weight}
                         1+(1+q)x=B_2(x,q).
\end{equation}

To pass inductively from \(\Lambda_{d-1}\) to \(\Lambda_d\) for \(d\geq 1\),
identify \(\mod\Lambda_{d-1}\) with the full subcategory of \(\mod\Lambda_d\)
consisting of modules supported on \(\{1,\ldots,d-1\}\).  The
\emph{restriction} of \((\TT',\FF')\in\PT(\Lambda_d)\) is
\[
 (\TT,\FF)=(\TT'\cap\mod\Lambda_{d-1},\,\FF'\cap\mod\Lambda_{d-1}).
\]
It is a pretorsion theory over \(\Lambda_{d-1}\), and we call
\((\TT',\FF')\) an \emph{extension} of \((\TT,\FF)\).  To describe all
extensions of a fixed \((\TT,\FF)\in\PT(\Lambda_{d-1})\) to \(\Lambda_d\),
let
\[
                         p_1<\cdots<p_k
\]
be the rows \(a\in\{1,\ldots,d-1\}\) for which \(\M{p_i}{d-1}\in\TT\), so
that \(k=\rho_{d-1}(\TT)\).  In cutoff coordinates, restricting a pretorsion
theory \((\Tvec{\beta},\Fvec{\alpha})\in\PT(\Lambda_d)\) to
\(\mod\Lambda_{d-1}\) deletes \(\beta_d\) and \(\alpha_d\) and replaces each
\(\beta_a\) (\(1\leq a\leq d-1\)) by \(\min(\beta_a,d-1)\), by~\eqref{eq:T-beta}
and~\eqref{eq:F-alpha}.  Because each row of a
quotient-closed subcategory is a prefix, any extension
\((\Tvec{\beta},\Fvec{\alpha})\) of \((\TT,\FF)\) satisfies
\(E_d(\beta)\subseteq\{p_1,\ldots,p_k,d\}\).  We say that row \(p_i\) is
\emph{continued to column \(d\)} when \(\beta_{p_i}=d\), that is, when
\(\M{p_i}{d}\in\Tvec{\beta}\).

\begin{lemma}\label{lem:restriction-extension}
Restriction defines a surjection \(\PT(\Lambda_d)\to\PT(\Lambda_{d-1})\).
The extensions of a pretorsion theory \((\TT,\FF)\in\PT(\Lambda_{d-1})\) to
\(\Lambda_d\) correspond bijectively to the choices of which rows among
\(p_1,\ldots,p_k\) are continued to column \(d\), whether \(\M{d}{d}\) is
included in the first component, and a value of \(\alpha_d\) permitted
by~\eqref{eq:allowed-alpha}.  Moreover, the indecomposable modules in the
intersection of an extension are those in \(\ind(\TT\cap\FF)\) together with
those in column \(d\) belonging to both new components.
\end{lemma}

\begin{proof}
For any module \(X\) supported on \(\{1,\ldots,d-1\}\), its largest
\(\TT'\)-submodule and its largest \(\FF'\)-quotient are also supported on
\(\{1,\ldots,d-1\}\), so its trace and reject with respect to
\((\TT',\FF')\) agree with those with respect to the restricted components
\((\TT,\FF)\), which remain quotient-closed and submodule-closed.
Restriction therefore preserves~\eqref{eq:trace-reject} and maps
\(\PT(\Lambda_d)\) into \(\PT(\Lambda_{d-1})\).  For any column \(j<d\),
continuing a row \(p_i\) from cutoff \(d-1\) to \(d\) changes neither the
condition \(\beta_a\geq j\) nor the condition \(\beta_a=j-1\), and therefore
leaves both \(E_j(\beta)\) and \(s_j(\beta)\) unchanged.  The new row
\(d\) also changes neither \(E_j(\beta)\) nor \(s_j(\beta)\), since \(j<d\).  By
Theorem~\ref{thm:staircase-pair}, a pair
\((\Tvec{\beta},\Fvec{\alpha})\in\LQ(\Lambda_d)\times\LS(\Lambda_d)\)
restricting to \((\TT,\FF)\) belongs to \(\PT(\Lambda_d)\) if and only if
\(\alpha_d\) satisfies~\eqref{eq:allowed-alpha}.  In particular, continuing
no row \(p_i\), omitting \(\M{d}{d}\), and setting \(\alpha_d=1\) yields an
extension of \((\TT,\FF)\), so restriction is surjective.  Distinct choices
of continued rows, of \(\M{d}{d}\), and of \(\alpha_d\) produce distinct
cutoff coordinates \((\beta,\alpha)\) and hence distinct pretorsion
theories.  Finally, for \(j<d\), we have \(\beta_a\geq j\) if and only if
\(\min(\beta_a,d-1)\geq j\), while \(\alpha_j\) is unchanged, so a module
\(\M{a}{j}\) with \(j<d\) belongs to \(\Tvec{\beta}\cap\Fvec{\alpha}\) if
and only if it belongs to \(\TT\cap\FF\).
\end{proof}

We enumerate extensions for arbitrary \(k=\rho_{d-1}(\TT)\geq0\), using the \(q\)-integers
\begin{equation}\label{eq:q-integer}
                         \qint r=1+q+\cdots+q^{r-1}\qquad(r\geq1).
\end{equation}
Fix \((\TT,\FF)\in\PT(\Lambda_{d-1})\) with \(\rho_{d-1}(\TT)=k\), and sum
over all extensions \((\TT',\FF')\in\PT(\Lambda_d)\) of \((\TT,\FF)\):
\begin{equation}\label{eq:F-k-def}
 \Phi_{\TT,\FF}(x,q)=\sum_{(\TT',\FF')}
 x^{\rho_d(\TT')}
 q^{|\ind(\TT'\cap\FF')|-|\ind(\TT\cap\FF)|}.
\end{equation}
By Lemma~\ref{lem:restriction-extension}, the exponent
\(\lvert\ind(\TT'\cap\FF')\rvert-\lvert\ind(\TT\cap\FF)\rvert\) is the
number of modules in column \(d\) belonging to \(\TT'\cap\FF'\), given
by~\eqref{eq:column-middle-size}.

We record two elementary identities used in the recurrences of this section
and Section~\ref{sec:image-closed}.  For an indeterminate \(w\), put
\(U=1+qw\) and let \(\Delta_q\) act on \(w\).  By the binomial theorem,
\begin{equation}\label{eq:binomial-u}
 \sum_{v=0}^{r}\binom{r}{v}(qw)^{v}
 =\sum_{v=0}^{r}\binom{r}{v}(qw)^{r-v}
 =U^{r}
 \qquad(r\geq0),
\end{equation}
and since \(U-w=1+(q-1)w\) is the denominator of \(\Delta_q\)
in~\eqref{eq:intro-delta}, the geometric sum formula gives
\begin{equation}\label{eq:geometric-delta}
 \sum_{i=0}^{\ell}w^{i}U^{\ell-i}
 =\frac{U^{\ell+1}-w^{\ell+1}}{U-w}
 =\Delta_q\bigl(w^{\ell+1}\bigr)
 \qquad(\ell\geq0).
\end{equation}

The next lemma evaluates \(\Phi_{\TT,\FF}(x,q)\) in terms of \(\Delta_q\).  In
\cite[p.~132, Lemma~13]{HanZeng}, Han and Zeng obtain a recurrence by
summing weights over the fibres of a restriction map;
we use the same method for the restriction of pretorsion theories from
\(\Lambda_d\) to \(\Lambda_{d-1}\).

\begin{lemma}\label{lem:first-transfer}
For \((\TT,\FF)\in\PT(\Lambda_{d-1})\), the sum
\(\Phi_{\TT,\FF}(x,q)\) in~\eqref{eq:F-k-def} depends only on
\(k=\rho_{d-1}(\TT)\).  Writing \(\Phi_k(x,q)\) for this common value, we have
\begin{equation}\label{eq:F-k-delta}
                         \Phi_k(x,q)=\Delta_q(x^{k+2}).
\end{equation}
Consequently, if
\[
 f(x)=\sum_{(\TT,\FF)\in\PT(\Lambda_{d-1})}x^{\rho_{d-1}(\TT)}q^{|\ind(\TT\cap\FF)|},
\]
then
\begin{equation}\label{eq:first-transfer-sum}
 \sum_{(\TT,\FF)\in\PT(\Lambda_d)}x^{\rho_d(\TT)}q^{|\ind(\TT\cap\FF)|}
 =\Delta_q\bigl(x^2f(x)\bigr).
\end{equation}
\end{lemma}

\begin{proof}
Fix \((\TT,\FF)\) and put \(k=\rho_{d-1}(\TT)\).  We first compute
\(\Phi_{\TT,\FF}(x,q)\) by summing over all extensions in column
\(d\), and then simplify the resulting expression.

\medskip
\noindent\textbf{Step 1: Counting the extensions.}
We compute \(\Phi_{\TT,\FF}(x,q)\) by partitioning the extensions according to the
first row among \(p_1,\ldots,p_k\) that is not continued to column \(d\).
First, suppose that all \(k\) rows \(p_1,\ldots,p_k\) are continued to
column \(d\).  If \(\M{d}{d}\) is added to \(\TT'\), then no row has cutoff
\(d-1\), so \(s_d(\beta)=d+1\); if \(\M{d}{d}\) is not added, then row \(d\)
is the first row with cutoff \(d-1\), so \(s_d(\beta)=d\).  In either case,
every row \(e\in E_d(\beta)\) satisfies \(e<s_d(\beta)\), so
\(\alpha_d=e+1\) is allowed by~\eqref{eq:allowed-alpha}.  When \(\M{d}{d}\) is added, \(\lvert E_d(\beta)\rvert=k+1\)
and the \(k+2\) choices of \(\alpha_d\) yield \(k+1,k,\ldots,0\) modules in
column \(d\) of \(\TT'\cap\FF'\), contributing \(x^{k+1}\qint{k+2}\); when
\(\M{d}{d}\) is not added, \(\lvert E_d(\beta)\rvert=k\) and the \(k+1\)
choices of \(\alpha_d\) contribute \(x^k\qint{k+1}\).

Second, suppose that \(p_j\) (\(1\leq j\leq k\)) is the first row among
\(p_1,\ldots,p_k\) that is not continued to column \(d\), and that \(v\) of
the \(k-j\) later rows \(p_{j+1},\ldots,p_k\) are also not continued to
column \(d\); there are \(\binom{k-j}{v}\) ways to choose these \(v\) rows.
Here \(s_d(\beta)=p_j\), so the permitted values of \(\alpha_d\)
in~\eqref{eq:allowed-alpha} are \(\alpha_d=1\) and \(\alpha_d=e+1\) for the
continued rows \(e=p_1,\ldots,p_{j-1}\) preceding \(p_j\).  If \(\M{d}{d}\)
is not added, then \(\lvert E_d(\beta)\rvert=k-1-v\).  The \(k-j-v\)
continued rows \(a>p_j\) satisfy \(a>e\) for every allowed choice
\(\alpha_d=e+1\) and therefore belong to \(\TT'\cap\FF'\) for every choice of \(\alpha_d\), contributing a
factor of \(q^{k-j-v}\), while the \(j\) choices of \(\alpha_d\) leave
\(j-1,j-2,\ldots,0\) of the rows \(p_1,\ldots,p_{j-1}\) in
\(\TT'\cap\FF'\), contributing \(\qint j\).  Thus the contribution when
\(\M{d}{d}\) is not added is
\[
                    \binom{k-j}{v}q^{k-j-v}\qint j x^{k-1-v}.
\]
When \(\M{d}{d}\) is added, \(d>p_i\) for every \(i<j\), so
\(\M{d}{d}\) also belongs to both components for every choice of
\(\alpha_d\).  This increases both \(\rho_d(\TT')\) and the number
of modules in column \(d\) of \(\TT'\cap\FF'\) by \(1\), contributing
\[
                    \binom{k-j}{v}q^{k-j-v+1}\qint j x^{k-v}.
\]
The resulting sum depends only on \(k\), proving the first assertion.
Denoting it by \(\Phi_k(x,q)\), we obtain
\begin{align}
 \Phi_k(x,q)
 &=x^{k+1}\qint{k+2}+x^k\qint{k+1}\notag\\
 &\quad+
 \sum_{j=1}^k\sum_{v=0}^{k-j}\binom{k-j}{v}
 q^{k-j-v}\qint j\bigl(x^{k-1-v}+qx^{k-v}\bigr).
 \label{eq:F-k-expanded}
\end{align}

\medskip
\noindent\textbf{Step 2: Evaluating the sum.}
Put \(U=1+qx\).  In the double sum in~\eqref{eq:F-k-expanded}, we factor
\(x^{k-1-v}+qx^{k-v}=Ux^{j-1}x^{k-j-v}\) and apply~\eqref{eq:binomial-u}
with \(w=x\) and \(r=k-j\) to obtain
\begin{align*}
 &\sum_{j=1}^k\qint j(1+qx)
   \sum_{v=0}^{k-j}\binom{k-j}{v}q^{k-j-v}x^{k-1-v}\\
 &\hspace{35mm}=
 U\sum_{j=1}^k\qint j x^{j-1}U^{k-j}.
\end{align*}
Consequently,
\begin{equation}\label{eq:F-k-single-sum}
 \Phi_k=x^{k+1}\qint{k+2}+x^k\qint{k+1}
       +U\sum_{j=1}^k\qint j x^{j-1}U^{k-j}.
\end{equation}
For \(k=0\), equation~\eqref{eq:F-k-single-sum} gives
\(\Phi_0=1+(1+q)x\).  For \(k\geq 1\), subtracting \(U\Phi_{k-1}\)
from~\eqref{eq:F-k-single-sum} cancels the terms with \(1\leq j<k\) in the
sum as well as the \(j=k\) term \(Ux^{k-1}\qint k\), leaving
\begin{align*}
 \Phi_k-U\Phi_{k-1}
 &=x^{k+1}\qint{k+2}+x^k\qint{k+1}
   -Ux^k\qint{k+1}\\
 &=x^{k+1}\bigl(\qint{k+2}-q\qint{k+1}\bigr)
  =x^{k+1}.
\end{align*}
Thus \(\Phi_k\) satisfies the recurrence
\begin{equation}\label{eq:F-k-recurrence}
                         \Phi_k=U\Phi_{k-1}+x^{k+1}\qquad(k\geq 1).
\end{equation}
On the other hand, by~\eqref{eq:geometric-delta},
\(\Delta_q(x^{k+2})=(U^{k+2}-x^{k+2})/(U-x)\) equals
\(1+(1+q)x\) for \(k=0\) and satisfies the same recurrence:
\[
 U\cdot\frac{U^{k+1}-x^{k+1}}{U-x}+x^{k+1}
 =\frac{U^{k+2}-Ux^{k+1}+x^{k+1}(U-x)}{U-x}
 =\frac{U^{k+2}-x^{k+2}}{U-x}.
\]
Hence \(\Phi_k(x,q)=\Delta_q(x^{k+2})\) for every \(k\geq 0\).

By Lemma~\ref{lem:restriction-extension}, every pretorsion theory
\((\TT',\FF')\in\PT(\Lambda_d)\) is an extension of a unique
\((\TT,\FF)\in\PT(\Lambda_{d-1})\).  Summing
\(q^{|\ind(\TT\cap\FF)|}\Phi_{\rho_{d-1}(\TT)}(x,q)\) over all
\((\TT,\FF)\in\PT(\Lambda_{d-1})\) and using~\eqref{eq:F-k-delta} together
with the \(\ZZ[q]\)-linearity of \(\Delta_q\) yields~\eqref{eq:first-transfer-sum}.
\end{proof}

\begin{example}\label{ex:B3}
By Lemma~\ref{lem:first-transfer} and~\eqref{eq:A1-pretorsion-weight},
the weighted count for \(n=2\) is
\(\Delta_q(x^2B_2(x,q))=B_3(x,q)\).  Applying~\eqref{eq:geometric-delta}
with \(U=1+qx\) gives
\[
 \Delta_q(x^2)=U+x=1+(1+q)x,
 \qquad
 \Delta_q(x^3)=U^2+Ux+x^2=1+(2q+1)x+(q^2+q+1)x^2,
\]
and hence
\[
 B_3(x,q)=\Delta_q\bigl(x^2+(1+q)x^3\bigr)
        =(2+q)+2(1+q)^2x+(1+q)(q^2+q+1)x^2.
\]
Setting \(x=1\) yields
\[
                         B_3(1,q)=5+7q+4q^2+q^3,
\]
so among the \(17\) pretorsion theories in \(\mod\Lambda_2\), there are
\(5\) torsion pairs (intersection size \(0\)), \(7\) pretorsion theories
with intersection of size \(1\), \(4\) with intersection of size \(2\), and
\(1\) with intersection of size \(3\), as can also be read from Table~\ref{tab:A2-complete}.
\end{example}

Iterating Lemma~\ref{lem:first-transfer} identifies the weighted generating
polynomial of \(\PT(\Lambda_n)\) with the \(q\)-Gandhi polynomial of the
first kind.

\begin{theorem}\label{thm:pretorsion-gandhi}
Let \(B_m(x,q)\) be the polynomials of~\eqref{eq:intro-B}.  For every
\(n\geq0\),
\begin{equation}\label{eq:pretorsion-D}
 \sum_{(\TT,\FF)\in\PT(\Lambda_n)}
 x^{\rho_n(\TT)}q^{|\ind(\TT\cap\FF)|}
                       =B_{n+1}(x,q).
\end{equation}
In particular,
\begin{equation}\label{eq:pretorsion-B}
 \sum_{(\TT,\FF)\in\PT(\Lambda_n)}q^{|\ind(\TT\cap\FF)|}
                       =B_{n+1}(1,q).
\end{equation}
\end{theorem}

\begin{proof}
We proceed by induction on \(n\geq 0\).  For \(n=0\), we have
\(\mod\Lambda_0=0\), so \(\PT(\Lambda_0)\) consists of the single pair
\((0,0)\), which has weight \(1=B_1(x,q)\) by
Definition~\ref{def:pretorsion-statistics}.  For \(n\geq 1\),
Lemma~\ref{lem:first-transfer} with \(d=n\) and the induction hypothesis
show that the left-hand side of~\eqref{eq:pretorsion-D} equals
\(\Delta_q(x^2 B_n(x,q))=B_{n+1}(x,q)\) by~\eqref{eq:intro-B}.  Evaluating
\eqref{eq:pretorsion-D} at \(x=1\) gives~\eqref{eq:pretorsion-B}.
\end{proof}

\begin{remark}\label{rem:Catalan-coefficient}
By Proposition~\ref{prop:zero-middle-torsion}, the constant term of
\(B_{n+1}(1,q)\) counts the torsion pairs in \(\mod\Lambda_n\).  By Corollary~\ref{cor:torsion-sublattice}, these pairs are in bijection
with torsion classes.  Thomas \cite[section ``Torsion classes in
\(\operatorname{rep} A_n\)'']{ThomasTamari} identifies their poset with the
Tamari lattice on bracketings of \(n+2\) letters, so this constant term is the Catalan number
\(\mathrm{Cat}_{n+1}=\frac{1}{n+2}\binom{2n+2}{n+1}\).  At the opposite
extreme, the coefficient of \(q^{n(n+1)/2}\) is \(1\), contributed only by
\((\mod\Lambda_n,\mod\Lambda_n)\).  Unlike size on \(\LI(\Lambda_n)\)
(Corollary~\ref{cor:LI-structure}(3)), however, the intersection size
\(\lvert\ind(\TT\cap\FF)\rvert\) is not a rank function on the lattice
\(\PT(\Lambda_n)\): already for \(n=1\), the three pretorsion theories of
Example~\ref{ex:A1-pretorsion} form a chain
\((0,\mod\Lambda_1)<(\mod\Lambda_1,\mod\Lambda_1)<(\mod\Lambda_1,0)\) in the
order~\eqref{eq:PT-order}, with intersection sizes \(0,1,0\).
Table~\ref{tab:coefficients} lists the coefficients of \(B_{n+1}(1,q)\) and
of the rank-generating polynomial \(C_{n+1}(1,q)\) from
Theorem~\ref{thm:image-gandhi} for \(n\leq4\).
\end{remark}

\begin{table}[ht]
\centering\small
\caption{Coefficients in ascending powers of \(q\).}\label{tab:coefficients}
\begin{tabular}{c p{.43\textwidth} p{.43\textwidth}}
\toprule \(n\)&\(B_{n+1}(1,q)\)&\(C_{n+1}(1,q)\)\\\midrule
0 & 1 & 1\\[3pt]
1 & 2, 1 & 1, 1\\[3pt]
2 & 5, 7, 4, 1 & 1, 3, 3, 1\\[3pt]
3 & 14, 36, 45, 35, 18, 6, 1 & 1, 6, 14, 17, 12, 5, 1\\[3pt]
4 & 42, 165, 330, 440, 433, 330, 198, 93, 33, 8, 1 & 1, 10, 40, 90, 132, 137, 105, 60, 25, 7, 1\\[3pt]
\bottomrule\end{tabular}
\end{table}

The identity \(B_m(1,1)=g_{2m+2}\), conjectured by Gandhi \cite{Gandhi},
was proved by Carlitz \cite{Carlitz} and by Riordan and Stein
\cite{RiordanStein}; see also Barsky \cite[Section~1, Lemma~1]{Barsky} and
Han and Zeng \cite[p.~120, Equation~(1)]{HanZeng} for the indexing
used here.  Thus~\eqref{eq:pretorsion-B} at \(q=1\) gives
\begin{equation}\label{eq:pretorsion-genocchi}
                         |\PT(\Lambda_n)|=g_{2n+4}.
\end{equation}
\section{Enumeration of image-closed subcategories by adjoining vertices}
\label{sec:image-closed}

In this section, we prove the second identity of
Theorem~\ref{thm:intro-enumeration} (Theorem~\ref{thm:image-gandhi}) by
adjoining one vertex at a time, parallel to Section~\ref{sec:first-gandhi}.

Identifying \(\mod\Lambda_{d-1}\) with the subcategory of
\(\Lambda_d\)-modules supported on \(\{1,\ldots,d-1\}\), we define the
\emph{restriction} of a subcategory \(\CC'\subseteq\mod\Lambda_d\) to be
\(\CC'\cap\mod\Lambda_{d-1}\).  Since the image of any morphism between
modules supported on \(\{1,\ldots,d-1\}\) is again supported on
\(\{1,\ldots,d-1\}\), restriction maps \(\LI(\Lambda_d)\) into
\(\LI(\Lambda_{d-1})\).  An \emph{extension} of
\(\CC\in\LI(\Lambda_{d-1})\) to \(\Lambda_d\) is an image-closed subcategory
\(\CC'\in\LI(\Lambda_d)\) whose restriction to \(\mod\Lambda_{d-1}\) is
\(\CC\); it is obtained by adjoining a subset of the indecomposable modules
\(\M{a}{d}\) (\(1\leq a\leq d\)) in column \(d\).

By Theorem~\ref{thm:intersection-criterion}, we may work with the
intersection-closed families of intervals in
Definition~\ref{def:interval-family}.  Whether a new interval \([a,d]\) can
be adjoined to an intersection-closed family \(\DD\) in
\(\{1,\ldots,d-1\}\) without generating new intervals in
\(\{1,\ldots,d-1\}\) is governed by the following condition on the left
endpoint \(a\).

\begin{definition}\label{def:admissible-starts}
Let \(\DD\) be an intersection-closed family of intervals in
\(\{1,\ldots,d-1\}\).  Define its set of
\emph{admissible starts for vertex \(d\)} by
\begin{equation}\label{eq:admissible-starts}
 \Adm_d(\DD)=\{a\in\{1,\ldots,d\}:
 [b,\ell]\in\DD,\ b\leq a\leq \ell\Longrightarrow[a,\ell]\in\DD\}.
\end{equation}
Thus \(a\in\Adm_d(\DD)\) if and only if every interval in \(\DD\) containing
\(a\) has its suffix beginning at \(a\) in \(\DD\).  In particular,
\(d\in\Adm_d(\DD)\) always holds vacuously because no interval in \(\DD\)
contains \(d\).
\end{definition}

The following example illustrates how an inadmissible start arises.

\begin{example}\label{ex:admissible-starts}
Let \(d=3\).  For the intersection-closed family
\[
                         \DD=\{[1,2]\}
\]
in \(\{1,2\}\), we have \(\Adm_3(\DD)=\{1,3\}\): the start \(a=2\) is not
admissible because \(2\in[1,2]\in\DD\) whereas \([2,2]\notin\DD\).  Thus one
may adjoin \([1,3]\), \([3,3]\), or both while preserving intersection
closure and keeping the restriction to \(\{1,2\}\) equal to \(\DD\), whereas
adjoining \([2,3]\) would require the intersection
\[
                         [1,2]\cap[2,3]=[2,2]
\]
to belong to the family, altering its restriction to \(\{1,2\}\).  By
contrast, for \(\DD=\{[1,2],[2,2]\}\), all three starts \(1,2,3\) are
admissible.
\end{example}

\subsection{Extensions of an intersection-closed interval family}

We first parametrize all subcategories of \(\mod\Lambda_d\) with a prescribed
restriction to \(\mod\Lambda_{d-1}\), and then characterize which of them
are image-closed and compute the admissible starts of the resulting family.

\begin{construction}\label{con:image-extension}
Let \(\CC\in\LI(\Lambda_{d-1})\), put \(\DD=\II(\CC)\), and choose a subset
\(J\subseteq\{1,\ldots,d\}\).  Define a subcategory
\(\CC_J\subseteq\mod\Lambda_d\) by
\begin{equation}\label{eq:D-J}
 \II(\CC_J)=\DD_J
   =\DD\cup\{[a,d]:a\in J\}.
\end{equation}
Every subcategory of \(\mod\Lambda_d\) whose restriction to
\(\mod\Lambda_{d-1}\) is \(\CC\) has the form \(\CC_J\) for a unique subset
\(J\subseteq\{1,\ldots,d\}\).
\end{construction}

The next lemma characterizes these extensions and computes the admissible
starts for the next vertex.  This update will let us iterate the count
using only the number of admissible starts.

\begin{lemma}\label{lem:admissible-extension}
Let \(\DD\) be an intersection-closed family in \(\{1,\ldots,d-1\}\)
and write \(\Adm_d(\DD)=\{c_1<\cdots<c_k\}\).
\begin{enumerate}
\item \(\DD_J\) is intersection-closed if and only if
      \(J\subseteq \Adm_d(\DD)\).
\item Suppose \(J\subseteq \Adm_d(\DD)\).  If \(J=\varnothing\), then
      \begin{equation}\label{eq:admissible-update-empty}
                  \Adm_{d+1}(\DD_J)=\Adm_d(\DD)\cup\{d+1\}.
      \end{equation}
      If \(J\neq\varnothing\) and \(\min J=c_j\), then
      \begin{equation}\label{eq:admissible-update-nonempty}
       \Adm_{d+1}(\DD_J)=\{c_1,\ldots,c_j\}
         \cup\bigl(J\cap\{c_{j+1},\ldots,c_k\}\bigr)
         \cup\{d+1\}.
      \end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
(1) The nonempty intersection of two intervals in \(\DD\) already belongs to
\(\DD\), and the intersection of two intervals \([a,d]\) and \([a',d]\)
ending at \(d\) is \([\max(a,a'),d]\), which belongs to \(\DD_J\) whenever
\(a,a'\in J\).  For \([b,\ell]\in\DD\) and \([a,d]\) with \(a\in J\), their
intersection is nonempty if and only if \(a\leq \ell\), and when \(a\leq b\) it
equals \([b,\ell]\in\DD\); when \(b\leq a\leq \ell\), it equals
\[
                       [b,\ell]\cap[a,d]=[a,\ell],
\]
which belongs to \(\DD_J\) if and only if \([a,\ell]\in\DD\).  Thus \(\DD_J\)
is intersection-closed if and only if every \(a\in J\) satisfies the
condition in~\eqref{eq:admissible-starts}, that is,
\(J\subseteq\Adm_d(\DD)\).

(2) Recall that \(c_k=d\in\Adm_d(\DD)\).  If
\(a\in\{1,\ldots,d\}\setminus\Adm_d(\DD)\), there exists
\([b,\ell]\in\DD\subseteq\DD_J\) with \(b\leq a\leq \ell\) and
\([a,\ell]\notin\DD\), and since \(\ell\leq d-1\) we still have
\([a,\ell]\notin\DD_J\); hence \(a\notin\Adm_{d+1}(\DD_J)\), so
\(\Adm_{d+1}(\DD_J)\subseteq\Adm_d(\DD)\cup\{d+1\}\).  If \(J=\varnothing\),
then \(\DD_J=\DD\) and every \(c_i\in\Adm_d(\DD)\) remains admissible for
vertex \(d+1\), proving~\eqref{eq:admissible-update-empty}.

Suppose now that \(J\neq\varnothing\) and \(\min J=c_j\).  Every old
interval containing \(c_i\) already has its suffix beginning at \(c_i\)
in \(\DD\subseteq\DD_J\), since \(c_i\in\Adm_d(\DD)\).  It remains to
check the new intervals.  For \(1\leq i\leq j\),
any new interval \([a,d]\) (\(a\in J\)) containing \(c_i\) satisfies
\(a\leq c_i\leq c_j=\min J\leq a\), so \(a=c_i\) and the suffix
\([c_i,d]\) belongs to \(\DD_J\); thus \(c_1,\ldots,c_j\in\Adm_{d+1}(\DD_J)\).
For \(j<i\leq k\), the new interval \([c_j,d]\in\DD_J\) contains \(c_i\), so
\(c_i\) belongs to \(\Adm_{d+1}(\DD_J)\) if and only if its suffix
\([c_i,d]\) belongs to \(\DD_J\), that is, \(c_i\in J\).  Finally, \(d+1\)
belongs to \(\Adm_{d+1}(\DD_J)\) vacuously.  This proves~\eqref{eq:admissible-update-nonempty}.
\end{proof}

Figure~\ref{fig:admissible-update} illustrates the update rule
\eqref{eq:admissible-update-nonempty}: any admissible start \(c_i\) after
\(c_j=\min J\) that is not included in \(J\) is no longer admissible at the
next step.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=11mm,y=8mm,>=Latex,font=\small]
 \foreach \i in {1,...,5}{
   \node[circle,draw,minimum size=5.5mm] (s\i) at (\i,0) {\(c_\i\)};
 }
 \node[above=3mm of s3] {\(\Adm_d(\DD)\)};
 \node[below=3mm of s2,blue!65!black] {first chosen};
 \node[below=3mm of s4,blue!65!black] {chosen};
 \draw[very thick,blue!65] (s2.south)--++(0,-.25);
 \draw[very thick,blue!65] (s4.south)--++(0,-.25);
 \draw[very thick,red!65] ($(s3)+(-.16,.16)$)--($(s3)+(.16,-.16)$);
 \draw[very thick,red!65] ($(s3)+(-.16,-.16)$)--($(s3)+(.16,.16)$);
 \draw[very thick,red!65] ($(s5)+(-.16,.16)$)--($(s5)+(.16,-.16)$);
 \draw[very thick,red!65] ($(s5)+(-.16,-.16)$)--($(s5)+(.16,.16)$);
 \draw[->] (6,0)--(7,0);
 \foreach \x/\lab in {1/c_1,2/c_2,4/c_4,6/{d+1}}{
   \node[circle,draw,fill=blue!8,minimum size=6mm] (t\x) at ({\x+7},0) {\(\lab\)};
 }
 \node[above=3mm of t4] {\(\Adm_{d+1}(\DD_J)\)};
\end{tikzpicture}
\caption{The update of admissible starts when \(J=\{c_2,c_4\}\) and
\(c_5=d\).  Red crosses mark starts that cease to be admissible.}
\label{fig:admissible-update}
\end{figure}

\subsection{A recurrence counting intersection-closed families}

We enumerate interval families by their size and number of admissible
starts, then recover the statistic \(\rho_n\) by substitution.

For \(m\geq 0\), define
\begin{equation}\label{eq:admissible-polynomial}
 Q_m(z,q)=\sum_{\DD}q^{|\DD|}z^{|\Adm_{m+1}(\DD)|},
\end{equation}
where the sum ranges over all intersection-closed families \(\DD\) of
intervals in \(\{1,\ldots,m\}\).  For \(m=0\), the only family is
\(\DD=\varnothing\), which has \(\Adm_1(\varnothing)=\{1\}\), so
\(Q_0(z,q)=z\).  Adjoining vertex \(n\) to families on
\(\{1,\ldots,n-1\}\) will express the generating polynomial for
\(\LI(\Lambda_n)\) as \(Q_{n-1}(1+qx,q)=C_{n+1}(x,q)\).

Fix \(d\geq 1\) and an intersection-closed family \(\DD\) in
\(\{1,\ldots,d-1\}\) with \(\lvert\Adm_d(\DD)\rvert=k\).  By
Lemma~\ref{lem:admissible-extension}, the extensions \(\DD_J\) of \(\DD\) to
\(\{1,\ldots,d\}\) correspond to subsets \(J\subseteq\Adm_d(\DD)\).  The
choice \(J=\varnothing\) has \(\lvert\DD_{\varnothing}\rvert=\lvert\DD\rvert\)
and \(\lvert\Adm_{d+1}(\DD_{\varnothing})\rvert=k+1\), contributing
\(q^{|\DD|}z^{k+1}\) to \(Q_d(z,q)\).  For \(J\neq\varnothing\), let
\(\min J=c_j\) (\(1\leq j\leq k\)) and choose \(v\) of the \(k-j\) later
elements \(c_{j+1},\ldots,c_k\) to belong to \(J\); there are
\(\binom{k-j}{v}\) such subsets \(J\), each having \(\lvert J\rvert=1+v\)
and \(\lvert\Adm_{d+1}(\DD_J)\rvert=j+v+1\) by~\eqref{eq:admissible-update-nonempty}.
Summing over all \(J\subseteq\Adm_d(\DD)\) shows that the extensions of
\(\DD\) contribute \(q^{|\DD|}\) times
\begin{align}
 z^{k+1}
  +\sum_{j=1}^k\sum_{v=0}^{k-j}
      \binom{k-j}{v}q^{1+v}z^{j+v+1}
 &=z^{k+1}+qz^2\sum_{j=1}^kz^{j-1}(1+qz)^{k-j}\notag\\
 &=z^{k+1}+qz^2\Delta_q(z^k),
 \label{eq:admissible-monomial-transfer}
\end{align}
where the first equality uses~\eqref{eq:binomial-u} with \(w=z\) and
\(r=k-j\), and the second uses~\eqref{eq:geometric-delta} with \(w=z\) and
\(\ell=k-1\) (with \(\Delta_q\) acting on \(z\)).  Summing
\eqref{eq:admissible-monomial-transfer} over all \(\DD\) in
\(\{1,\ldots,d-1\}\) gives the recurrence
\begin{equation}\label{eq:admissible-transfer}
            Q_d(z,q)=zQ_{d-1}(z,q)+qz^2\Delta_qQ_{d-1}(z,q).
\end{equation}

Recall from Definition~\ref{def:pretorsion-statistics} that \(\rho_n(\CC)\)
counts the indecomposable modules in \(\CC\) whose support ends at \(n\).
By Construction~\ref{con:image-extension},
Lemma~\ref{lem:admissible-extension}(1), and
Theorem~\ref{thm:intersection-criterion}, every
\(\CC'\in\LI(\Lambda_n)\) with \(n\geq1\) is of the form \(\CC_J\)
for a unique \(\CC\in\LI(\Lambda_{n-1})\) and a unique subset
\(J\subseteq\Adm_n(\II(\CC))\), with \(\rho_n(\CC_J)=\lvert J\rvert\) and
\(\lvert\ind\CC_J\rvert=\lvert\ind\CC\rvert+\lvert J\rvert\).  Summing over
\(\CC\in\LI(\Lambda_{n-1})\) and \(J\subseteq\Adm_n(\II(\CC))\) therefore
yields
\[
 \sum_{\CC'\in\LI(\Lambda_n)}x^{\rho_n(\CC')}q^{|\ind\CC'|}
 =\sum_{\CC\in\LI(\Lambda_{n-1})}q^{|\ind\CC|}
   \sum_{J\subseteq \Adm_n(\II(\CC))}(qx)^{|J|}.
\]
Since the inner sum equals \((1+qx)^{|\Adm_n(\II(\CC))|}\), comparing with
\eqref{eq:admissible-polynomial} gives
\begin{equation}\label{eq:image-from-P}
 \sum_{\CC\in\LI(\Lambda_n)}x^{\rho_n(\CC)}q^{|\ind\CC|}
                         =Q_{n-1}(1+qx,q).
\end{equation}

In the proof of the following theorem, we verify that substituting
\(z=1+qx\) in~\eqref{eq:admissible-transfer} gives the defining
recurrence~\eqref{eq:intro-C} for the \(q\)-Gandhi polynomials of the
second kind.

\begin{theorem}\label{thm:image-gandhi}
Let \(C_m(x,q)\) be defined by~\eqref{eq:intro-C}.
For every \(n\geq0\),
\begin{equation}\label{eq:image-C}
 \sum_{\CC\in\LI(\Lambda_n)}x^{\rho_n(\CC)}q^{|\ind\CC|}
                         =C_{n+1}(x,q).
\end{equation}
In particular, the rank-generating polynomial of the graded lattice
\(\LI(\Lambda_n)\) is
\begin{equation}\label{eq:image-C-at-one}
 \sum_{\CC\in\LI(\Lambda_n)}q^{|\ind\CC|}=C_{n+1}(1,q).
\end{equation}
\end{theorem}

\begin{proof}
We first prove by induction on \(d\geq 1\) that
\begin{equation}\label{eq:P-C-induction}
                         Q_{d-1}(1+qx,q)=C_{d+1}(x,q).
\end{equation}
For \(d=1\), both sides equal \(1+qx\) since \(Q_0(z,q)=z\) and
\(C_2(x,q)=(1+qx)\Delta_q(x)=1+qx\).

For the induction step, assume~\eqref{eq:P-C-induction} for \(d\), and put
\(U=1+qx\) and \(V=1+(q-1)x=U-x\).  Applying~\eqref{eq:P-C-induction} at
\(x\) and at \(U\) gives
\[
 Q_{d-1}(U,q)=C_{d+1}(x,q),
 \qquad
 Q_{d-1}(1+qU,q)=C_{d+1}(U,q).
\]
Since \(1+(q-1)U=qV\), evaluating \(\Delta_q Q_{d-1}(z,q)\) at \(z=U\) yields
\[
 q\,\bigl(\Delta_qQ_{d-1}\bigr)(U,q)
 =q\,\frac{Q_{d-1}(1+qU,q)-Q_{d-1}(U,q)}{1+(q-1)U}
 =\frac{C_{d+1}(U,q)-C_{d+1}(x,q)}{V}.
\]
Substituting \(z=U\) into~\eqref{eq:admissible-transfer} and using
\(V-U=-x\), we obtain
\begin{align}
 Q_d(U,q)
 &=UC_{d+1}(x,q)\notag
   +U^2\frac{C_{d+1}(U,q)-C_{d+1}(x,q)}{V}\notag\\
 &=\frac{U^2C_{d+1}(U,q)-UxC_{d+1}(x,q)}{V}\label{eq:image-recurrence-substitution}\\
 &=U\Delta_q\bigl(xC_{d+1}(x,q)\bigr)
  =C_{d+2}(x,q),\notag
\end{align}
completing the induction for~\eqref{eq:P-C-induction}.

For \(n\geq 1\), combining~\eqref{eq:image-from-P} with~\eqref{eq:P-C-induction}
at \(d=n\) gives~\eqref{eq:image-C}; for \(n=0\), both sides
of~\eqref{eq:image-C} equal \(1\).  Setting \(x=1\)
gives~\eqref{eq:image-C-at-one}, where \(\lvert\ind\CC\rvert\) is the rank
function of \(\LI(\Lambda_n)\) by Corollary~\ref{cor:LI-structure}(3).
\end{proof}

Grouping pretorsion theories by their intersection \(\ZZZ=\TT\cap\FF\)
relates \(B_{n+1}(1,q)\) and \(C_{n+1}(1,q)\) through the number
\(\lvert\PT_{\ZZZ}(\Lambda_n)\rvert\) of realizations of each
\(\ZZZ\in\LI(\Lambda_n)\):
\[
 B_{n+1}(1,q)=\sum_{\ZZZ\in\LI(\Lambda_n)}
       \lvert\PT_{\ZZZ}(\Lambda_n)\rvert\,q^{|\ind\ZZZ|},
 \qquad
 C_{n+1}(1,q)=\sum_{\ZZZ\in\LI(\Lambda_n)}q^{|\ind\ZZZ|}.
\]
In Table~\ref{tab:A2-complete}, for instance, the eight image-closed
subcategories for \(n=2\) have \(5,3,1,3,1,2,1,1\) realizations.

We also check Theorem~\ref{thm:image-gandhi} directly for \(n=2\),
recovering the count of Example~\ref{ex:A2-image}.

\begin{example}\label{ex:C3}
By Example~\ref{ex:A2-image}, every subset of the three intervals
\([1,1],[1,2],[2,2]\) in \(\{1,2\}\) is intersection-closed.  Because
\([1,2]\) and \([2,2]\) end at vertex \(2\) while \([1,1]\) does not, choosing
each interval independently yields
\[
 \sum_{\CC\in\LI(\Lambda_2)}x^{\rho_2(\CC)}q^{|\ind\CC|}
                   =(1+q)(1+qx)^2.
\]
From the recurrence~\eqref{eq:intro-C}, we have \(C_2(x,q)=1+qx\) and,
using \(\Delta_q(x^2)=1+(1+q)x\),
\[
 C_3(x,q)=(1+qx)\Delta_q\bigl(x+qx^2\bigr)
        =(1+qx)\bigl(1+q+q(1+q)x\bigr)
        =(1+q)(1+qx)^2.
\]
At \(x=1\), this specializes to \(C_3(1,q)=(1+q)^3\).
\end{example}

Using the classical identity \(C_m(1,1)=H_{2m+1}\)
\cite[pp.~63--64, Equation~(2)]{HanZengMedian},
equation~\eqref{eq:image-C-at-one} at \(q=1\) gives
\begin{equation}\label{eq:image-median}
                         |\LI(\Lambda_n)|=H_{2n+3}.
\end{equation}

\begin{remark}\label{rem:normalized-image}
Han and Zeng \cite[Section~4, Equation~(17)]{HanZengMedian} prove that
\((1+q)^n\) divides \(C_{n+1}(1,q)\) and define the polynomial
\(\bar c_{n+1}(q)=C_{n+1}(1,q)/(1+q)^n\).
Feigin \cite[Corollary~2.4]{FeiginGenocchi} identifies
\(\bar c_{n+1}(q)=q^{n(n-1)/2}h_n(q^{-1})\), where \(h_n\) is the
Poincar\'e polynomial of the degenerate complete flag variety for
\(\mathbb C^n\) (with cohomological degrees divided by two), and Cerulli
Irelli, Feigin, and Reineke \cite[Proposition~2.7]{CFR} realize these
degenerate flag varieties as quiver Grassmannians for linearly oriented quivers of
type~\(A\).
\end{remark}


\begin{remark}\label{rem:orientation-counts}
The counts of pretorsion theories and image-closed subcategories depend on
the orientation of the quiver.  The table below was computed over
\(\mathbb F_2\) from the Hom spaces and images of interval modules by
testing Theorem~\ref{thm:trace-reject} on every pair of a quotient-closed
and a submodule-closed subcategory.  By Theorem~\ref{thm:image-middle}, the
distinct intersections of these pretorsion theories give the
\(\lvert\LI\rvert\) column, while by
Proposition~\ref{prop:zero-middle-torsion} the pairs with zero
intersection give the column of torsion pairs.
The resulting counts hold over every
field, for the reason given below.  Each word in \(L\) and \(R\) records the
orientation of the arrows between consecutive vertices \(1,\ldots,n\), with
\(R\) denoting \(i\to i+1\) and \(L\) denoting \(i\leftarrow i+1\);
orientations with equal counts are grouped together.
\begin{center}
\begin{tabular}{c l r r r}
\toprule
\(n\)&Orientations&\(|\PT|\)&\(|\LI|\)&Torsion pairs\\
\midrule
1&empty word&3&2&2\\
2&\(L,R\)&17&8&5\\
3&\(LL,RR\)&155&56&14\\
 &\(LR,RL\)&163&64&14\\
4&\(LLL,RRR\)&2073&608&42\\
 &\(LLR,LRR,RLL,RRL\)&2297&784&42\\
 &\(LRL,RLR\)&2441&880&42\\
\bottomrule
\end{tabular}
\end{center}
For any orientation of \(A_n\) and any base field, the indecomposable
modules are the thin interval representations, with a one-dimensional
vector space at each vertex of their support and identity maps along the
arrows of the support.  A nonzero morphism between two such
representations is a scalar multiple of the identity on the overlap of
their supports; whether such a morphism exists depends only on the arrow
orientations, and its image and kernel are coordinate subspaces.
Consequently, the trace and the reject of any set of indecomposables,
evaluated on an interval module, are sums and intersections of fixed
coordinate subspaces, and both they and the resulting submodules and
quotients (which decompose into thin interval modules) are independent of
the field.  By Proposition~\ref{prop:trace-reject-properties}(3),
membership in the quotient closure of a set is detected by whether its
trace is the whole module, and membership in its submodule closure by
whether its reject vanishes; moreover,
Proposition~\ref{prop:trace-reject-properties}(4) reduces the
trace--reject equalities on arbitrary modules to those on interval modules.
Thus the quotient-closed and
submodule-closed subcategories, the pairs satisfying
Theorem~\ref{thm:trace-reject}, and—by Theorem~\ref{thm:image-middle}—their
intersections, the image-closed subcategories, are all independent of the
field.

Extending the \(\mathbb F_2\) computation to all orientations with
\(n\leq 6\) shows that, in this finite range, the linearly oriented quivers
have the smallest numbers of pretorsion theories and image-closed
subcategories, whereas the alternating orientations have the largest.  In
contrast, for every \(n\) and every orientation of \(A_n\), the number of
torsion pairs is \(\mathrm{Cat}_{n+1}\): over an algebraically closed field
this follows from the bijection between torsion classes and noncrossing
partitions \cite[Theorem~1.1]{IngallsThomas}, and the argument above showing
independence of the field extends the count to arbitrary fields.
\end{remark}

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\end{document}