Abstract
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.
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Acknowledgements
We thank the referees for helpful comments. The first author was supported by JSPS KAKENHI Grant No. 22K20343 and Grant-in-Aid for JSPS Research Fellow, Grant No. JP19J22605, Japan. The second author was supported by JST PRESTO Grant Number JPMJPR192A, Japan. No datasets were generated or analysed during the current study.
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Hayashi, K., Hirai, H. Two Flags in a Semimodular Lattice Generate an Antimatroid. Order (2023). https://doi.org/10.1007/s11083-023-09639-5
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DOI: https://doi.org/10.1007/s11083-023-09639-5