Weak Uninorm Based Logic and Its Filter Theory

Abstract

We give an axiomatic system of a logic called here a weak uninorm based logic (wUL), which is proved to be characterized by the class of all (not necessary bounded nor integral) commutative residuated lattices. We see that the logic is algebraizable. Since many well-known logics, e.g., UBL by Watari and al., UL by Metcalfe and Montanga, ML by H\"ohle, MTL by Esteva and L. Godo, BL by H\'ajek, and so on, are axiomatic extensions of our logic, those logics are all algebraizable. Moreover we define filters of commutative residuated lattices $X$ and show that the class of all filters of $X$ is isomorphic to the class $Con(X)$ of all congruences on $X$. At last, as an application of our characterization of wUL, we give a negative answer to the problem that "Is UBL characterized by the class of linearly ordered UBL-algebras?", which was left open in [10].