On Skew Heyting Algebras

TL;DR

This paper extends Heyting algebras to a non-commutative framework, defining skew Heyting algebras and providing various examples to illustrate their structure and applications.

Contribution

It introduces the concept of skew Heyting algebras, generalizing Heyting algebras to non-commutative settings, and offers multiple examples including dual skew Boolean algebras.

Findings

Introduction of skew Heyting algebras as a new algebraic structure

Examples include Heyting algebras, dual skew Boolean algebras, and algebras of partial maps

Establishes the proper notion of implication in skew lattices

Abstract

In the present paper we generalize the notion of a Heyting algebra to the non-commutative setting and hence introduce what we believe to be the proper notion of the implication in skew lattices. We list several examples of skew Heyting algebras, including Heyting algebras, dual skew Boolean algebras, conormal skew chains and algebras of partial maps with poset domains.