On ideals of a skew lattice

TL;DR

This paper reviews the concept of ideals in skew lattices, exploring their properties, relationships with lattice images, and their connection to the coset structure, advancing the understanding of skew lattice order theory.

Contribution

It provides a comprehensive review of ideals in skew lattices and examines their intersection with coset structures, highlighting their unique properties and relationships.

Findings

Analysis of ideals derived from preorder and partial order

Clarification of the relationship between skew ideals and lattice images

Discussion of the intersection of ideals with coset structures

Abstract

Ideals are one of the main topics of interest to the study of the order structure of an algebra. Due to their nice properties, ideals have an important role both in lattice theory and semigroup theory. Two natural concepts of ideal can be derived, respectively, from the two concepts of order that arise in the context of skew lattices. The correspondence between the ideals of a skew lattice, derived from the preorder, and the ideals of its respective lattice image is clear. Though, skew ideals, derived from the partial order, seem to be closer to the specific nature of skew lattices. In this paper we review ideals in skew lattices and discuss the intersection of this with the study of the coset structure of a skew lattice.