Additive generator pairs of overlap functions

Abstract

Let $\theta:[0,1]\rightarrow[-\infty,+\infty]$ be a function with both $\theta(x^{-})$ and $\theta(x^{+})$ existing for every $x\in [0,1]$ and $\vartheta:[-\infty,+\infty]\rightarrow[-\infty,+\infty]$ be a function. In this article we completely characterize the pair $(\theta,\vartheta)$ for the bivariate function $O_{\theta,\vartheta}: [0,1]^{2}\rightarrow[0,1]$ given by $$O_{\theta,\vartheta}(x,y)=\vartheta(\theta(x)+\theta(y))$$ being an overlap function. In particular, we give analytical expressions of some transformations for the pair $(\theta,\vartheta)$.