On the Finite Embeddability Property for Residuated Lattices, Pocrims and BCK-algebras

To see that a finitely axiomatizable logic is decidable—in the sense that it has a decidable set of theorems—it suffices to show that it has the finite model property, and indeed, this has been the method of choice for establishing decidability of propositional logics. A logic is said to have the finite model property (FMP) if every formula that fails to be a theorem of the logic can be refuted in a finite model of the logic. The first one to apply the method in a non-trivial way was J.C.C. McKinsey in [6], where he obtained decision procedures for the modal logics S2 and S4. Although traditionally logics have often been identified with their sets of theorems, work on the algebraization of logic (such as in [1]) has emphasized the importance of the inferences of the logic. We say that a logic has