Maximal Sublattices of Convex Geometries with Unbounded Complement Minima
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.
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