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Fuzzy implicative and Boolean ideals of \(MV\)-algebras. (English) Zbl 0844.06007

Summary: We show how to obtain results in \(MV\)-algebras by considering their fuzzy ideals. In particular we prove the following theorem: Let \(\mu\) be a fuzzy ideal of an \(MV\)-algebra \(X\). Then the following are equivalent: (1) \(\mu(x\wedge\overline x)=\mu(0)\) for all \(x\in X\), (2) \(\mu(x^n)=\mu(x)\) for all \(x\in X\) and all \(n\geq 1\), (3) \(X/X_\mu\) is a Boolean algebra, (4) \(\mu\) is fuzzy implicative. If \(I\) is an ideal of \(X\) and we take \(\mu=\chi_I\) then we obtain corresponding results for the ideal \(I\) and \(X\). We show how to extend fuzzy prime and fuzzy implicative ideals. Various fuzzy radicals are considered and related to the corresponding radicals of the algebra, and we show how they live in the spectrum of the algebra.

MSC:

06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
06B10 Lattice ideals, congruence relations
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