Additive generator pairs of overlap functions
Li-zhi Liang, Xue-ping Wang
TL;DR
The paper addresses when a bivariate function of the form $O_{\theta,\vartheta}(x,y)=\vartheta(\theta(x)+\theta(y))$ is an overlap function by characterizing the additive generator pair $(\theta,\vartheta)$. It establishes precise continuity, monotonicity, and boundary conditions for $\theta$ and $\vartheta$, and links a threshold parameter $a$ to the behavior of the generators. It then develops transformation rules, showing that affine and other transformations preserve the additive generator structure, and provides concrete examples and domain-extensions for generating new overlap functions. Together, these results offer a systematic framework for constructing and transforming overlap functions via one-variable generators, with potential applications in fuzzy logic and related fields.
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 $(θ,\vartheta)$.