Flat Coset Decompositions of Skew Lattices

TL;DR

This paper explores flat coset decompositions in skew lattices, providing insights into their structure by analyzing how skew lattices can be broken down into minimal building blocks through cosets.

Contribution

It introduces the concept of flat coset decompositions in skew lattices and examines their role in understanding the fundamental structure of these non-commutative lattices.

Findings

Skew lattices can be decomposed into flat cosets.

Flat coset decompositions reveal minimal structural components.

The approach generalizes classical lattice decompositions.

Abstract

Skew lattices are non-commutative generalizations of lattices, and the cosets represent the building blocks that skew lattices are built of. As by Leech's Second Decomposition Theorem any skew lattice embeds into a direct product of a left-handed skew lattice by a right-handed one, it is natural to consider the so called flat coset decompositions, i.e. decompositions of a skew lattice into right and left cosets, thus finding the smallest atoms that compose the structure.