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

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.