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arXiv · 2608.17438

Algebras with straightening laws on join- or meet-semidistributive lattices

Koji Matsushita·Sora Miyashita·Koichiro Tani
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Abstract

We study algebras with straightening laws on join- or meet-semidistributive lattices. We show that for a join-semidistributive (resp. meet-semidistributive) lattice, meet-distributivity (resp. join-distributivity) and Cohen--Macaulayness are equivalent, and that integrality implies these conditions. Thus, Hibi's conjecture that every integral lattice is Cohen--Macaulay holds for join- or meet-semidistributive lattices. For semidistributive lattices, distributivity, integrality and Cohen--Macaulayness are equivalent.

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Koji Matsushita, Sora Miyashita, Koichiro Tani. 2026-08-18. Algebras with straightening laws on join- or meet-semidistributive lattices. https://arxiv.org/abs/2608.17438

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