Convergence in MV-algebras
George Georgescu, Fortunata Liguori, Giulia Martini · RACO (Revistes Catalanes amb Accés Obert) (Consorci de Serveis Universitaris de Catalunya) · 1997
$MV$-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.