Interpolation and FEP for Logics of Residuated Algebras

A residuated algebra (RA) is a generalization of a residuated groupoid; instead of one basic binary operation · with residual operations \, /, it admits finitely many basic operations, and each n−ary basic operation is associated with n residual operations. A basic logical system for RAs was studied in e.g. [6, 8, 16, 15] under the name: Generalized Lambek Calculus GL. In this paper we study GL and its extensions in the form of sequent systems. We prove an interpolation property which allows to replace a substructure of the antecedent structure by a single formula in a provable sequent. Together with model constructions, based on nuclei [13], interpolation leads to proofs of Finite Embeddability Property of different classes of RAs, as e.g. all RAs, distributive lattice-ordered RAs, boolean RAs, Heyting RAs and double RAs.