On the equivalence between MV-algebras and l-groups with strong unit

EDUARDO J. DUBUC AND YURI POVEDA

STUDIA LOGICA

Springer

Año: 2014

0039-3215

This note concerns the category $mathcal{M}$ of MV-algebras and its full subcategory $mathcal{CM}$ of totally ordered MV-algebras, for short, mbox{emph{MV-chains},} and the category $mathcal{G}_u$ of abelian l-groups with strong unit, for short mbox{emph{elu -groups},} with unital l-homomorphism as arrows, and its full subcategory $mathcal{CG}_u$ of totally ordered abelian l-groups with strong unit, for short mbox{emph{elu-chains}.} In cite{Ch} C.C. Chang proved that any $MV$-chain $A$ was isomorphic to the segment $A cong Gamma(A^*, u)$ of a elu-chain $A^*$. He constructs $A^*$ by the simple intuitive idea of putting denumerable copies of $A$ on top of each other (indexed by the integers). Moreover, he also shows that any elu-chain $G$ can be recovered from its segment, since $G cong Gamma(G, u)^*$, establishing an equivalence of categories.In cite{M} D. Mundici extended this result to arbitrary $MV$-algebras and elu-groups. He takes a representation of $A$ as a sub-direct product of chains $A_i$, and observes that $A mmr{} prod_i A_i^*$. Then he let $A^*$ be the subgroup generated by $A$ inside $prod_i A_i^*$. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of emph{good sequences} and its complicated arithmetics. In this note, essentially self-contained except for Chang´s result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the product $l$-group $prod_i A_i^*$, avoiding entirely the notion of good sequence.