Two Flags in a Semimodular Lattice Generate an Antimatroid
2024
Online
Elektronische Ressource
A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown (Abels 1991, Herscovici 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.
Online Published: 13 June 2023
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Two Flags in a Semimodular Lattice Generate an Antimatroid
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Veröffentlichung: | 2024 |
Medientyp: | Elektronische Ressource |
DOI: | 10.1007/s11083-023-09639-5 |
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