The logic of the strongest and the weakest t-norms

Abstract

It is well known that the strongest t-norm, that is the largest with respect to the pointwise order, is the minimum. In 2001, the logic MTL was introduced as the base of a framework of many-valued logics, and in 2002 it was shown that it is the logic of all left-continuous t-norms and their residua. Within this family of logics, the many-valued logic associated with the minimum t-norm is the Gödel one, whilst there is no logic associated to the drastic product t-norm. Indeed the drastic product is not left-continuous, and hence it does not have a residuum. However, in a recent paper the logic DP has been studied, by showing that the monoidal operation of every DP-chain is like the drastic product t-norm. In this paper we present the logic EMTL, whose algebraic variety is the smallest to contain the ones of Gödel- and DP-algebras. We show that the chains in this algebraic variety are exactly all the Gödel- and DP-chains, we classify and axiomatize all the subvarieties, and we show some limitative results concerning the amalgamation property.