An Algebraic Approach to Propositional Fuzzy Logic

Abstract

We investigate the variety corresponding to a logic (introduced in Esteva and Godo, 1998, and called LΠ there), which is the combination of Lukasiewicz Logic and Product Logic, and in which Godel Logic is interpretable. We present an alternative (and slightly simpler) axiomatization of such variety. We also investigate the variety, called the variety of LΠ½ algebras, corresponding to the logic obtained from LΠ by the adding of a constant and of a defining axiom for one half. We also connect LΠ½ algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that every LΠ½ algebra \cal A can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield \cal F, and whose operations are the truncations of the operations of \cal F to [0, 1]. We prove that such a structure \cal F is uniquely determined by \cal A up to isomorphism, and we establish an equivalence between the category of LΠ½ algebras and that of f-semifields.