2014/08/05 by Eduardo J. Dubuc, Dubuc, Eduardo J., Yuri A. Poveda +1
Computer Science · Decision Sciences · #Advanced Algebra and Logic #FOS: Mathematics #Fuzzy and Soft Set Theory #Logic (math.LO) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.1408.1070
openalex publication_date 2014/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≅ 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. 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 Ai, and observes that A \overset \hookrightarrow ∏i Gi where Gi = Ai^*. 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.