An Overview of Migrative Triangular Norms

In this paper we summarize our results obtained recently on continuous triangular norms that are migrative with respect to an arbitrary continuous triangular norm. We start with the original notion of migrativity and completely describe all continuous migrative triangular norms. Since the migrative property excludes both idempotent and nilpotent t-norm classes, the characterization and construction is carried out by solving a functional equation for additive generators of strict t-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, to reach our main goal, we exploit the ordinal sum structure of continuous t-norms and our former results related to the migrative property. We illustrate the statements by numerical examples and figures. Key–Words: continuous triangular norm, migrative property, functional equations, extended migrative property, minimum t-norm, product t-norm, Łukasiewicz t-norm, ordinal sum.