On a class of partially ordered sets and their linear invariants

Let (ℒ, <) be a finite partially ordered set with rank function. Then ℒ is the disjoint union of the classes ℒ k of elements of rank k and the order relation between elements in ℒ k and ℒ k+1 can be represented by a matrix S k. We study partially ordered sets which satisfy linear recurrence relations of the type S k (S k T ) − c k (S k − 1)T S k − 1 = d k + − c k d k −) Id for all k and certain coefficients d k +, d k - and c k.