Distributive lattices determined by weighted double skeletons

TL;DR

This paper characterizes finite distributive lattices using weighted double skeletons, showing that if the second skeleton is trivial, the lattice is uniquely determined by this structure.

Contribution

It introduces a new characterization of finite distributive lattices via weighted double skeletons, emphasizing the importance of the second skeleton's triviality.

Findings

Finite distributive lattices are uniquely determined by their weighted double skeletons when the second skeleton is trivial.

The second skeleton's triviality is a crucial condition for this characterization.

The work extends the understanding of lattice structures through skeleton-based invariants.

Abstract

Related to his S-glued sum construction, the skeleton S(L) of a finite lattice L was introduced by C. Herrmann in 1973. Our theorem asserts that if D is a finite distributive lattice and its second skeleton, S(S(D)), is the trivial lattice, then D is characterized by its weighted double skeleton, introduced by the second author in 2006. The assumption on the second skeleton is essential.