Recent results on migrative triangular norms

Abstract

This paper summarizes our recent results on continuous triangular norms that are migrative with respect to an arbitrary continuous triangular norm. The content is based on three journal papers [5], [7], [8]. We start with the original notion of migrativity and completely describe all continuous migrative triangular norms. Then we extend the migrative property by allowing an arbitrary but fixed t-norm in the defining equation instead of the originally used product t-norm. Equivalent forms of this extended migrativity are also provided. Two particular cases when the fixed t-norm is either the minimum or the Łukasiewicz t-norm are studied. In these cases all continuous extended migrative t-norms are characterized and represented. Finally, we exploit the ordinal sum structure of continuous t-norms and our mentioned results to describe all continuous triangular norms that are migrative with respect to an arbitrary continuous triangular norm. We illustrate the statements by numerical examples.