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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.

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