Partially ordered sets of distributive type and algebras with straightening laws

TL;DR

This paper investigates finite posets of distributive type and their associated algebras with straightening laws, focusing on conditions for Cohen--Macaulayness and Gorenstein properties.

Contribution

It generalizes the join-meet toric ring to ASLs on finite posets of distributive type and studies their Cohen--Macaulay and Gorenstein conditions.

Findings

Identifies conditions under which these algebras are Cohen--Macaulay.

Provides criteria for Gorenstein property in these algebras.

Analyzes a natural class of finite posets of distributive type.

Abstract

A finite poset (partially ordered set) $P$ with ${\hat 0}$ is called of distributive type if every interval $[{\hat 0}, a]$, $a \in P$, of $P$ is a distributive lattice. From a viewpoint of ASL's (algebras with straightening laws), the join-meet toric ring on a finite distributive lattice is generalized to an ASL on a finite poset of distributive type. Our target is the questions when a finite poset of distributive lattice is Cohen--Macaulay and when the ASL on it is Gorenstein. We focus on a natural class of finite posets of distributive type and study various aspects of the above questions.