Additive generator pairs of overlap functions

Let $\theta :[0,1]\to [-\mathrm{\infty },+\mathrm{\infty }]$ be a function with both $\theta (x^{-})$ and $\theta (x^{+})$ existing for every $x\in [0,1]$ and $\vartheta :[-\mathrm{\infty },+\mathrm{\infty }]\to [-\mathrm{\infty },+\mathrm{\infty }]$ be a function. In this article we completely characterize the pair $(\theta ,\vartheta )$ for the bivariate function $O_{\theta ,\vartheta }:[0,1{]}^2\to [0,1]$ 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 $(\theta ,\vartheta )$.