An algebraic approach to propositional fuzzy logic

We investigate the variety corresponding to a logic (introduced in Esteva and Godo, 1998, and called Łπ there), which is the combination of Łukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative (and slightly simpler) axiomatization of such variety. We also investigate the variety, called the variety of Łπ1/2 algebras, corresponding to the logic obtained from Łπ by the adding of a constant and of a defining axiom for one half. We also connect Łπ1/2 algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that every Łπ1/2 algebra A can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield ℱ, and whose operations are the truncations of the operations of ℱ to [0, 1]. We prove that such a structure ℱ is uniquely determined by A up to isomorphism, and we establish an equivalence between the category of Łπ1/2 algebras and that of f-semifields