2022/04/07 by Koyo Hayashi, Hayashi, Koyo, Hiroshi Hirai +1
Computer Science · #Advanced Algebra and Logic #Rough Sets and Fuzzy Logic
paper · doi:10.48550/arxiv.2204.03188
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.