LP and LP1\2: two fuzzy logics joining Lukasiewicz and Product logics

In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called LΠ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from LΠ by the adding of a constant symbol and of a defining axiom for 1/2, called LΠ1/2. We show that LΠ1/2 contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with Δ, and the Product and Gödel's Logics with Δ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to LΠ and LΠ1/2. For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z.