Partially ordered sets of distributive type and algebras with straightening laws

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.

Figures (2)

• Figure 1: A poset of distributive type
• Figure 2: A poset with linear resolution

Theorems & Definitions (46)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Corollary 2.4
• Example 2.5
• Lemma 2.6
• proof
• Lemma 3.1
• proof
• Lemma 3.2