Orthocomplete Pseudo MV-algebras

TL;DR

This paper introduces the concept of orthocomplete pseudo MV-algebras, extending the theory of MV-algebras by using functorial methods to establish orthocompletions for representable pseudo MV-algebras.

Contribution

It defines orthocomplete pseudo MV-algebras and proves their existence for all representable pseudo MV-algebras using a generalized Mundici's functor.

Findings

Orthocomplete pseudo MV-algebras are introduced and characterized.

Every representable pseudo MV-algebra has an orthocompletion.

Results also apply to classical MV-algebras.

Abstract

Pseudo $MV$-algebras are a non-commutative generalization of $MV$-algebras. The main purpose of the paper is to introduce and investigate orthocomplete pseudo $MV$-algebras. We use the concepts of projectable pseudo $MV$-algebras and large pseudo $MV$-subalgebras to introduce orthocomplete pseudo $MV$-algebras. Then we apply a generalization of the Mundici's functor to an orthocompletion of an representable $\ell$-group to prove that each representable pseudo $MV$-algebra has an orthocompletion. In particular, our results are valid also for $MV$-algebras.