Convergence in MV-algebras

Abstract

$MV$-algebras were introduced in 1958 by Chang and they are models of Lukasiewicz infinite-valued logic. Chang gives a correspondence between the category of linearly ordered $MV$-algebras and the category of linearly ordered abelian $\ell$-groups.

Mundici extended this result showing a categorical equivalence between the category of the $MV$-algebras and the category of the abelian $\ell$-groups with strong unit.

In this paper, starting from some definitions and results in abelian $\ell$-groups, we shall study the convergent sequences and the Cauchy sequences in an $MV$-algebra.

The main result is the construction of the Cauchy completion $A^{*}$ of an $MV$-algebra $A$.

It is proved that a complete $MV$-algebra is also Cauchy complete.

Additional results on atomic and complete $MV$-algebras are also given.