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Noncommutative extensions in algebra of logic. (English) Zbl 1059.03081

Summary: BL algebras were introduced by P. Hájek as algebraic structures for his basic logic. MV algebras, product algebras and Gödel algebras are special cases of BL algebras. We present here some noncommutative extensions introduced in the last two years. Pseudo-MV algebras extend MV-algebras in the same way in which arbitrary \(l\)-groups extend Abelian \(l\)-groups. Pseudo-product algebras extend product algebras. Pseudo-BL algebras extend BL algebras and contain pseudo-MV algebras and pseudo-product algebras as special cases. There is no noncommutative extension of Gödel algebras. Pseudo-t-norms extend t-norms.

MSC:

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