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On weakly cancellative fuzzy logics. (English) Zbl 1113.03021
The paper deals with fuzzy logics over the monoidal t-norm based logic. It starts with the quite general result that for each MTL-chain (like for each BL-chain) there is a maximum decomposition as an ordinal sum of indecomposable totally ordered prelinear semihoops (with the first semihoop bounded). Unlike the situation in BL, where indecomposable components are completely characterized, the situation in MTL is still unclear. For this reason the authors focus on a particular class of indecomposable MTL-chains, namely weakly cancellative MTL-chains (WCMTL-chains), and their ordinal sums. Considering these chains and the corresponding fuzzy logics, the authors present the following results: First, an axiomatization of the logic of ordinal sums of WCMTL-chains is given. Second, it is proved that none of the considered varieties is locally finite and satisfies the finite model property and the finite embeddability property. Further, it is shown that all studied logics enjoy strong standard completeness for finite theories, but for none of them it is possible to extend this result to infinite theories. Finally, the predicate versions of the considered logics are discussed, and it is proved that they are not strong standard complete even for finite theories. More precisely, the authors prove that the finitary consequence relation w.r.t. the standard semantics is not recursively enumerable.

MSC:
03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
03G25 Other algebras related to logic
03G20 Logical aspects of Łukasiewicz and Post algebras
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