On EMV-algebras with square roots

TL;DR

This paper introduces and studies the concept of square roots on EMV-algebras, generalizing known properties from MV-algebras, and explores their structural implications and characterizations within algebraic frameworks.

Contribution

It extends the theory of square roots from MV-algebras to EMV-algebras, providing new properties, characterizations, and relations with representing structures.

Findings

Square roots characterize EMV-algebras.

Relation established between square roots on EMV-algebras and their top-element counterparts.

Complete characterization of square roots via group addition in unital ℓ-groups.

Abstract

A square root is a unary operation with some special properties. In the paper, we introduce and study square roots on EMV-algebras. First, the known properties of square roots defined on MV-algebras will be generalized for EMV-algebras, and we also find some new ones for MV-algebras. We use square roots to characterize EMV-algebras. Then, we find a relation between the square root of an EMV-algebra and the square root on its representing EMV-algebra with top element. We show that each strict EMV-algebra has a top element and we investigate the relation between divisible EMV-algebras and EMV-algebras with a special square root. Finally, we present square roots on tribes, EMV-tribes, and we present a complete characterization of any square root on an MV-algebra and on an EMV-algebra by group addition in the corresponding unital $\ell$-group.