Note on additive generator pairs of overlap and grouping functions

TL;DR

This paper investigates the relationships and conditions under which overlap and grouping functions are additively generated by pairs of functions, linking them to triangular norms and transformations.

Contribution

It characterizes when overlap functions are triangular norms and how they relate to distortions and automorphisms, extending understanding of additive generator pairs.

Findings

Conditions for overlap functions to be triangular norms.

Characterization of additive generator pairs via distortions and automorphisms.

Dual results for grouping functions.

Abstract

In this article, we deeply reveal the relationship between functions $\theta$ and $\vartheta$ in an overlap function additively generated by an additive generator pair ($\theta$,$\vartheta$). Then we characterize the conditions for an overlap function additively generated by the pair being a triangular norm by terms of functions $\theta$ 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.