Additive generator pairs of overlap functions

Abstract

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

12 pages