Putting together Łukasiewicz and product logics

Abstract

In this paper we investigate a propositional fuzzy logical system $\L\Pi$ which contains the well-known \L ukasiewicz, Product and G\"{o}del fuzzy logics as sublogics. We define the corresponding algebraic structures, called $\L\Pi$-algebras and prove the following completeness result: a formula $\varphi$ is provable in the $\L\Pi$ logic iff it is a tautology for all linear $\L\Pi$-algebras. Moreover, linear $\L\Pi$-algebras are shown to be embeddable in linearly ordered abelian rings with a strong unit and cancellation law.