Alternative Proof of Standard Completeness Theorem for MTL

Abstract

As it was shown in Jenei and Montagna (Studia Logica 70(2): 183---192, 2002), Monoidal t-norm based logic (MTL) is a logic of left-continuous t-norms. In other words, this means that MTL enjoys the standard completeness theorem. In this paper we present a different proof of this theorem. In fact, we prove even more since we show that MTL is complete w.r.t. the class of standard MTL-algebras with finite congruence lattice or equivalently with finitely many Archimedean classes. We also show the connection between the congruence lattice of an MTL-chain and its Archimedean classes.