Two flags in a semimodular lattice generate an antimatroid

TL;DR

This paper extends the understanding of how two flags in a semimodular lattice generate a structure, showing that their modular convex hull forms an antimatroid, linking lattice theory with combinatorial structures.

Contribution

It establishes an analogue of the two-flag generation theorem for semimodular lattices, connecting modular convex hulls with antimatroids and shortest galleries.

Findings

Modular convex hull of two flags is isomorphic to a union-closed family.

This family uniquely determines an antimatroid.

The join-sublattice of shortest galleries coincides with the antimatroid.

Abstract

A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown (Abels 1991, Herscovic 1998) that two flags in a semimodular lattice no longer generate such a good sublattice, whereas shortest galleries connecting them form a relatively good join-sublattice. In this note, we sharpen this investigation to establish an analogue of the two-flag generation theorem for a semimodular lattice. We consider the notion of a modular convex subset, which is a subset closed under the join and meet only for modular pairs, and show that the modular convex hull of two flags in a semimodular lattice of rank $n$ is isomorphic to a union-closed family on $[n]$. This family uniquely determines an antimatroid, which coincides with the join-sublattice of shortest galleries of the two flags.