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

Studia Logica 103 (4):807-814 (2015)

Abstract
In “A new proof of the completeness of the Lukasiewicz axioms” Chang proved that any totally ordered MV-algebra A was isomorphic to the segment \}\) 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. Moreover, he also show that any such group G can be recovered from its segment since \^*}\), establishing an equivalence of categories. In “Interpretation of AF C *-algebras in Lukasiewicz sentential calculus” 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 \ where \. Then he let A * be the l-subgroup generated by A inside \. 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 \, avoiding entirely the notion of good sequence
Keywords MV-Algebras   l-Groups  Good Sequences
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DOI 10.1007/s11225-014-9593-9
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