Addition of Sigmoid-Shaped Fuzzy Numbers Using the Dombi operator and Infinite Sum Theorems J. Dombi and N. Gy˝orb´ ır´o University of Szeged, Institute of Informatics {dombi | gyorbiro}@inf.u-szeged.hu Abstract The extension principle defines the arithmetic operations on fuzzy numbers. In the extension principle one can use any t-norm for mod- eling the conjunction operator. It is therefore important to know, which t-norms are consistent with a particular type of fuzzy number. 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. We prove that the addition is closed if the Dombi operator is used. Keywords: Zadeh’s extension principle, Dombi operator, sigmoid-shaped fuzzy number. 1 Introduction Many real world applications have to work with imprecise data. Results of measurements, vague statements, flexible constraints can all be a source of in- accurate information. Fuzzy quantities provide a mathematical model for such imprecise quantities and perceptions. The idea that fuzzy quantities could be arithmetically combined according to the laws of fuzzy set theory is due to Zadeh [25]. Soon after, several researchers worked independently along these lines, such as Jain [16], Mizumoto and Tanaka [21], [22], Nahmias [23], Nguyen [24], Dubois and Prade [4]. An overview of fuzzy arithmetics can be found in Dubois and Prade [6], [8], [1]. Several theoretical details and applications can be found e.g., in monographs of Kaufmann and Gupta [17], [18], and Mares [19]. Two special issues of Fuzzy Sets and Systems [7], [12] were also devoted to the topic. The computational approach to linguistic quantifiers is investigated by Zadeh [26], [27]. 1