Constructing/analyzing differential distributed lattices

TL;DR

This paper revisits a process for constructing and analyzing differential distributive lattices, extending it to weighted structures and aiming for a comprehensive characterization of such lattices.

Contribution

It extends Stanley's technique to weighted differential lattices and proposes using it as a basis for their complete characterization.

Findings

Proves the uniqueness of $d$-differential distributive lattices for any positive integer $d$.

Extends the process to weighted-differential lattices with positive weights.

Provides a framework for analyzing properties of these lattices.

Abstract

We restate a process presented by Stanley as a technique to prove that there exists exactly one $d$-differential distributive lattice for any positive integer $d$. This process can be trivially extended to apply to distributive finitary lattices that have a variety of differential poset structures. It can be viewed as an algorithm for constructing such lattices. Alternatively, it can be viewed as an algorithm for analyzing and characterizing such lattices. We show that the process can be used to prove properties of all weighted-differential lattices with positive weights. We present this with the hope that this approach can be used as the basis for a complete characterization of distributive lattices with a weighted-differential structure with positive weights.