Additive generator pairs of overlap functions

TL;DR

This paper characterizes the pairs of functions that generate overlap functions through additive combinations, providing analytical expressions and a complete description of such pairs.

Contribution

It offers a full characterization of additive generator pairs for overlap functions, including explicit transformations and analytical expressions.

Findings

Complete characterization of generator pairs for overlap functions

Analytical expressions for transformations of generator pairs

Conditions under which the sum of generators yields an overlap function

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 $(\theta,\vartheta)$.