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

In “A new proof of the completeness of the Lukasiewicz axioms”} (Transactions of the American Mathematical Society, 88) C.C. Chang proved that any totally ordered $MV$-algebra $A$ was isomorphic to the segment $A\cong \mathrm{\Gamma }(A^{\ast },u)$ of a totally ordered $l$-group with strong unit $A^{\ast }$. This was done by the simple intuitive idea of putting denumerable copies of $A$ on top of each other (indexed by the integers). Moreover, he also show that any such group $G$ can be recovered from its segment since $G\cong \mathrm{\Gamma }(G,u{)}^{\ast }$, establishing an equivalence of categories. In “Interpretation of AF $C^{\ast }$-algebras in Lukasiewicz sentential calculus” (J. Funct. Anal. Vol. 65) D. Mundici extended this result to arbitrary $MV$-algebras and $l$-groups with strong unit. He takes the representation of $A$ as a sub-direct product of chains $A_i$, and observes that $A{\prod }_i{G}_i$ where $G_i = A_i^$\mathrm{.}Thenhelet$A^$bethe$l$-subgroupgeneratedby$A$inside$\prod_i G_i$. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of 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 the product $l$-group $\prod_i G_i$, avoiding entirely the notion of good sequence.