×

zbMATH — the first resource for mathematics

Tensor products and the Loomis-Sikorski theorem for MV-algebras. (English) Zbl 0926.06004
The author defines the MV-tensor product of MV-algebras. However, since it is possible for a semisimple MV-algebra to have a tensor product with itself which is non-semisimple, he restricts himself to semisimple algebras and defines their semisimple tensor product, and gives a way to visualize it in terms of separating subalgebras of the algebra of continuous \([0,1]\)-valued functions on the set of maximal ideals. If the algebra is also what he calls multiplicative, he defines a natural product on it. He proves a generalization of the Loomis-Sikorski theorem, namely:
Theorem. Let \(A\) be a \(\sigma\)-complete MV-algebra and let \(X\) be the set of maximal ideals. Then there is a tribe \({\mathcal F}\) over \(X\) and a \(\sigma\)-homomorphism \(\eta\) of \({\mathcal F}\) onto \(A\). In fact, if \(A\) is also multiplicative, then \({\mathcal F}\) can be chosen to be closed under pointwise multiplication and \(\eta\) is a homomorphism from this to the natural multiplication.
Reviewer: C.S.Hoo (Edmonton)

MSC:
06D35 MV-algebras
46L05 General theory of \(C^*\)-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et anneaux Réticulés, Lecture notes in mathematics, 608, (1971), Springer-Verlag Berlin
[2] Chang, C.C., Algebraic analysis of many-valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704
[3] Chang, C.C., A new proof of the completeness of the łukasiewicz axioms, Trans. amer. math. soc., 93, 74-80, (1959) · Zbl 0093.01104
[4] Cignoli, R.; Mundici, D., An invitation to Chang’s MV-algebras, Advances in algebra and model theory, (1997), Gordon and Breach Publishing Group Reading, p. 171-197 · Zbl 0935.06010
[5] Cignoli, R.; D’Ottaviano, I.M.L.; Mundici, D., Algebras of łukasiewicz logics, Editions CLE, (1995), State University of Campinas Campinas
[6] Goodearl, K.R., Partially ordered abelian groups with interpolation, (1986), American Mathematical Society Providence · Zbl 0589.06008
[7] Mundici, D., Interpretation of AFC, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059
[8] Mundici, D., Free products in the category of abelian ℓ-groups with strong unit, J. algebra, 113, 89-109, (1988) · Zbl 0658.06010
[9] Riečan, B.; Neubrunn, T., Integral, measure, and ordering, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0916.28001
[10] Sikorski, R., Boolean algebras, (1960), Springer-Verlag Berlin · Zbl 0191.31505
[11] Yosida, K., Functional analysis, (1980), Springer-Verlag Berlin · Zbl 0152.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.