Quantifier elimination and other model-theoretic properties of BL-algebras
BL is short for Hájek Basic Logic, the logic of continuous t-norms and their adjoint implications. BL-algebras are the Lindenbaum algebras of theories in BL-logic, as described in \textit{P. Hájek}'s book [Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. When the underlying order of a BL-algebra \(A\) is total, \(A\) is said to be a BL-chain. It is easy to see that the theory of all BL-algebras does not have quantifier elimination. The present paper features a model-theoretic analysis of the first-order theory of classes of BL-chains, with particular reference to classes generating the whole variety and whose first-order theories have quantifier elimination. Applications are given to the amalgamation problem for various classes of BL-algebras. Ultrahomogeneous models and the joint embedding properties are also considered, along with decidability and completeness. Finally, the authors prove a sort of BL-algebraic variant of the MV-algebraic Di Nola theorem. This sort of model-theoretical analysis for MV-algebras (the algebras of Łukasiewicz logic) was carried out by Lacava and Marchioni. Caicedo proved quantifier elimination for the chains arising from rational Pavelka logic. Quantifier elimination results for Gödel and product algebras were proved by Baaz and Veith. In an earlier paper, Marchioni established quantifier for certain theories of totally ordered commutative residuated lattices and obtained the amalgamation property of various classes of chains by purely model-theoretic means.
scientific article; zbMATH DE number 5991004