Note on additive generator pairs of overlap and grouping functions

Abstract

In this article, we deeply reveal the relationship between functions $\hat{I}¸$ and $\vartheta$ in an overlap function additively generated by an additive generator pair ($\hat{I}¸$,$\vartheta$). Then we characterize the conditions for an overlap function additively generated by the pair being a triangular norm by terms of functions $\hat{I}¸$ and $\vartheta$. We also establish the conditions that an overlap function additively generated by the additive generator pair can be obtained by a distortion of a triangular norm and a (pseudo) automorphism. Finally, we dually give the related results concerned grouping functions.

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