New family of triangular norms via contrapositive symmetrization of residuated implications

Abstract

In this paper a certain transformation of triangular norms (called N-annihilation) is studied. This problem is strongly related to the contrapositive symmetry property of residuated implications. We characterize those continuous triangular norms where the annihilated binary operation is a triangular norm. Some surprising properties of nilpotent triangular norms are presented e.g. the nilpotent minimum is described as limit of nilpotent triangular norms. As a consequence, a new family of triangular norms (called nilpotent ordinal sums) owing several attractive properties is discovered. The new family contains the nilpotent triangular norms and the nilpotent minimum as extremal cases and can be admitted into investigations in the theory of Girard monoids as well.