On the coset category of a skew lattice

TL;DR

This paper explores the categorical structure of skew lattices, focusing on their coset decomposition, and identifies conditions under which skew lattices in rings form a well-defined category.

Contribution

It introduces the coset category of skew lattices, analyzes when skew lattices determine such categories, and provides conditions for matrix skew lattices to be distributive.

Findings

Not all skew lattices determine a coset category.

Examples of skew lattices in rings that are not categorical.

Conditions for matrix skew lattices to be $ abla$-distributive.

Abstract

Skew lattices are non-commutative generalizations of lattices. The coset structure decomposition is an original approach to the study of these algebras describing the relation between its rectangular classes. In this paper we will look at the category determined by these rectangular algebras and the morphisms between them, showing that not all skew lattices can determine such a category. Furthermore, we will present a class of examples of skew lattices in rings that are not strictly categorical, and present sufficient conditions for skew lattices of matrices in rings to constitute $\wedge$-distributive skew lattices.