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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
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