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 ∼= Γ(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 ∼= Γ(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 Ai, and observes that A → i Gi where Gi = A∗ i . Then he let A∗ be the l-subgroup generated by A inside i Gi. 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 i Gi, avoiding entirely the notion of good sequence.