Homology of Distributive Lattices

TL;DR

This paper develops a homology theory for distributive lattices and related structures, providing formulas for their homology and proposing generalizations of lattice concepts.

Contribution

It introduces multi-term distributive homology for sets with distributive operations and derives a complete homology formula for finite distributive lattices.

Findings

Complete homology formula for finite distributive lattices

Results on homology of unital spindles

Conjectures for semi-lattices and skew lattices

Abstract

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology of a finite distributive lattice. We also indicate the answer for unital spindles and conjecture the general formula for semi-lattices and some skew lattices. Then we propose a generalization of a lattice as a set with a number of idempotent operations satisfying the absorption law.