Y. T. Miah

math.CO — Combinatorics

Maximal Sublattices of Convex Geometries with Unbounded Complement Minima

Contributed by Y. T. Miah

For every integer $r\ge 3$, we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly $r$ minimal elements and exactly two maximal elements. The lattice has $(r+2)^2$ elements and the sublattice has $(r+1)(r+2)/2+3$ elements. At $r=3$ the construction is the 25-element example of Bhattacharjee. Thus the number of minimal elements in such a complement is unbounded even when the convex dimension is fixed at three, in contrast to the common-minimum theorem in convex dimension two.