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)


06D35 MV-algebras
46L05 General theory of \(C^*\)-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
Full Text: DOI


[1] Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et Anneaux Réticulés. Groupes et Anneaux Réticulés, Lecture Notes in Mathematics, 608 (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0384.06022
[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: 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. Algebras of Łukasiewicz Logics, Editions CLE (1995), State University of Campinas: State University of Campinas Campinas
[6] Goodearl, K. R., Partially Ordered Abelian Groups with Interpolation (1986), American Mathematical Society: American Mathematical Society Providence · Zbl 0589.06008
[7] Mundici, D., Interpretation of AF \(C\), 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: Kluwer Academic Publishers Dordrecht · Zbl 0916.28001
[10] Sikorski, R., Boolean Algebras (1960), Springer-Verlag: Springer-Verlag Berlin · Zbl 0191.31505
[11] Yosida, K., Functional Analysis (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0152.32102
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