On the Equivalence Between MV-Algebras and l-Groups with Strong Unit

Abstract

In “A new proof of the completeness of the Lukasiewicz axioms” (Trans Am Math Soc 88, 1959) Chang proved that any totally ordered MV-algebra A was isomorphic to the segment ${A \cong \Gamma(A^*, u)}$ of a totally ordered l-group with strong unit A ^*. 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 ${G \cong \Gamma(G, u)^*}$, establishing an equivalence of categories. In “Interpretation of AF C ^*-algebras in Lukasiewicz sentential calculus” (J Funct Anal 65, 1986) 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 ${A \hookrightarrow \prod_i G_i}$ where ${G_i = A_i^*}$. Then he let A ^* be the l-subgroup generated by 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.