On the coset structure of a skew lattice

TL;DR

This paper explores the algebraic structure of skew lattices, focusing on their coset decomposition and the relation between order and coset structures, providing insights into their internal algebraic organization.

Contribution

It introduces a new perspective on skew lattices by relating their order structure to coset decompositions and revisiting known algebraic decompositions.

Findings

Revealed the connection between order and coset structures in skew lattices.

Revisited and clarified known decompositions of skew lattices.

Highlighted the role of Green's relation D as a congruence in the algebraic category.

Abstract

The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of Sets where the Green's relation D is a congruence describing an adjunction to the category of Lattices. In this paper we will discuss the relevance of this approach, revisit some known decompositions and relate the order structure of a skew lattice with its coset structure that describes the internal coset decomposition of the respective algebra.