On the Equivalence Between MV-Algebras and l-Groups with Strong Unit
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 c...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2015 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/97915 |
| Acceso en línea: | http://hdl.handle.net/11336/97915 |
| Access Level: | acceso abierto |
| Palabra clave: | MV ALGEBRAS L GROUPS GOOD SEQUENCES https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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 AFC*-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↪∏iGi where Gi = Ai∗. Then he let A* be the l-subgroup generated by A inside ∏iGi. 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 ∏iGi, avoiding entirely the notion of good sequence. |
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