n-contractive BL-logics

In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in Hájek (Metamathematics of fuzzy logic, trends in logic. Kluwer Aca- demic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom φ^n → φ^{n+1}, for some n ∈ N^+) axiomatic extensions of BL and their corresponding varieties: BL^n, SBL^n, BL_n and SBL_n. Concerning BL^n we have that every BL^n-chain is isomorphic to an ordinal sum of MV-chains of at most n+1 elements, whilst every BL_n-chain is isomorphic to an ordinal sum of MV_n-chains (for SBL^n and SBL_n a similar property holds, with the difference that the first component must be the two elements boolean algebra); all these varieties are locally finite. Moving to the content of the paper, after a preliminary section, we will study generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we will analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.