On epicomplete $MV$-algebras

TL;DR

This paper investigates the properties of epicomplete $MV$-algebras, establishing their relation to divisible and injective $MV$-algebras, and introduces the concept of epicompletion, proving its universal existence for all $MV$-algebras.

Contribution

It characterizes epicomplete $MV$-algebras, links them to divisible and injective $MV$-algebras, and introduces the concept of epicompletion with existence results.

Findings

Epicomplete $MV$-algebras are equivalent to divisible $MV$-algebras.

A relation between injective and epicomplete $MV$-algebras is established.

Every $MV$-algebra admits an epicompletion.

Abstract

The aim of the paper is to study epicomplete objects in the category of $MV$-algebras. A relation between injective $MV$-algebras and epicomplete $MV$-algebras is found, an equivalence condition for an $MV$-algebra to be epicomplete is obtained, and it is shown that the class of divisible $MV$-algebras and the class of epicomplete $MV$-algebras are the same. Finally, the concept of an epicompletion for $MV$-algebras is introduced, and the conditions under which an $MV$-algebra has an epicompletion are obtained. As a result we show that each $MV$-algebra has an epicompletion.