AI 中文总结
本文研究半分配格上的直化代数，证明并-或半分配格中交/并分配性与Cohen--Macaulay性质等价，整性蕴含该性质，还对半分配格得出分配性、整性与Cohen--Macaulay性质等价的结论，验证了Hibi的相关猜想。
AI 中文摘要
我们研究并-或半分配格上的直化代数。证明对于并半分配格（对应地，交半分配格），交分配性（对应地，并分配性）与Cohen--Macaulay性质等价，且整性蕴含这些条件，故Hibi提出的“每个整格都是Cohen--Macaulay格”的猜想对并-或半分配格成立；对半分配格，分配性、整性与Cohen--Macaulay性质等价。
英文摘要
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.
Comments7 pages, 1 figure
