Further properties of reversible triangular norms

Abstract

In this paper we present some properties of discontinuous reversible triangular norms. Two of them seem to be important: (1) a reversible t-norm T which possesses only trivial idempotent elements must be Archimedean. (2) If T is a discontinuous reversible t-norm and has non-trivial idempotent elements, then T is an ordinal sum of two t-noms, the one on the lower part of the unit square dominating Lukasiewcz t-norm, the other being reversible and Archimedean. Consequently, no left-continuous t-norm that is not continuous is reversible. Therefore, the question of reversibility of discontinuous t-norms is reducible to the case when the t-norms in question possess trivial idempotent elements only. Moreover, we have completely characterized the reversibility of T * (the reverse of a reversible t-norm T).