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The prime and maximal spectra and the reticulation of BL-algebras. (English) Zbl 1039.03052
BL-algebras are the algebras of Hájek’s Basic Logic [see P. Hájek, Metamathematics of fuzzy logic. Trends in Logic – Studia Logica Library 4, Kluwer Academic Publishers, Dordrecht (1998; Zbl 0937.03030)]. In the paper under review the author investigates the prime and the maximal spectrum of BL algebras. Equipped with the usual hull-kernel topology, the former turns out to be a spectral space, and the latter a compact Hausdorff space. In the final part of the paper the authors gives the BL-algebraic counterpart of the reticulation functor, along the lines of the Simmons functor (for commutative rings) and of the Belluce functor (for noncommutative rings and MV-algebras). The reticulation functor sends every BL algebra \(A\) into a normal completely normal lattice \(B\) in such a way that the prime spectra of \(A\) and \(B\) are homeomorphic.

MSC:
03G25 Other algebras related to logic
06D35 MV-algebras
03B52 Fuzzy logic; logic of vagueness
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