Partially ordered sets of distributive type and algebras with straightening laws

A finite poset (partially ordered set) $P$ with $\widehat{0}$ is called of distributive type if every interval $[\widehat{0},a]$, $a\in P$, 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.