Addition of sigmoid-shaped fuzzy intervals using the Dombi operator and infinite sum theorems

The extension principle defines the arithmetic operations on fuzzy intervals. In the extension principle one can use any t-norm for modeling the conjunction operator. It is therefore important to know, which t-norms are consistent with a particular type of fuzzy intervals. We call a t-norm consistent, if the arithmetic operation is closed. In this paper we investigate the addition of sigmoid and two bell-shaped membership functions which appear in many natural processes and are used in machine learning applications. We prove that the addition is closed if the Dombi operator is used. The calculation of sum is quite simple and can be used in various applications such as fuzzy-neural networks.