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\(n\)-contractive BL-logics. (English) Zbl 1266.03041
Summary: In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in [P. Hájek, Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. In this paper we study four families of \(n\)-contractive (i.e., satisfying the axiom \({\varphi^n\rightarrow\varphi^{n+1}}\), for some \({n\in\mathbb{N}^+}\)) axiomatic extensions of BL and their corresponding varieties: \(\mathrm{BL}^{n}\), \(\mathrm{SBL}^{n}\), \(\mathrm{BL}_{n}\) and \(\mathrm{SBL}_{n}\). Concerning \(\mathrm{BL}^{n}\) we have that every \(\mathrm{BL}^{n}\)-chain is isomorphic to an ordinal sum of MV-chains of at most \(n + 1\) elements, whilst every \(\mathrm{BL}_{n}\)-chain is isomorphic to an ordinal sum of \(\mathrm{MV}_{n}\)-chains (for \(\mathrm{SBL}^{n}\) and \(\mathrm{SBL}_{n}\) a similar property holds, with the difference that the first component must be the two-element Boolean algebra); all these varieties are locally finite. After a preliminary section, we study generic and \(k\)-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.

03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
03G25 Other algebras related to logic
06B20 Varieties of lattices
06D35 MV-algebras
Full Text: DOI
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