Additive generator pairs of overlap functions

Abstract

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