Morphisms on $EMV$-algebras and Their Applications

TL;DR

This paper introduces $EMV$-morphisms for a new class of algebras called $EMV$-algebras, establishing their categorical properties, free constructions, and relations to $MV$-algebras, expanding the algebraic framework.

Contribution

It defines $EMV$-morphisms, studies their categorical structure, and constructs free and weakly free $EMV$-algebras, connecting them to existing $MV$-algebra theory.

Findings

Category of $EMV$-algebras is closed under product.

Existence of free $EMV$-algebras on finite sets.

Introduction of weakly free $EMV$-algebras for infinite sets.

Abstract

For a new class of algebras, called $EMV$-algebras, every idempotent element $a$ determines an $MV$-algebra which is important for the structure of the $EMV$-algebra. Therefore, instead of standard homomorphisms of $EMV$-algebras, we introduce $EMV$-morphisms as a family of $MV$-homomorphisms from $MV$-algebras $[0,a]$ into other ones. $EMV$-morphisms enable us to study categories of $EMV$-algebras where objects are $EMV$-algebras and morphisms are special classes of $EMV$-morphisms. The category is closed under product. In addition, we define free $EMV$-algebras on a set $X$ with respect to $EMV$-morphisms. If $X$ is finite, then the free $MV$-algebra on $X$ is a free $EMV$-algebras. For an infinite set $X$, the same is true introducing a so-called weakly free $EMV$-algebra.