Set-based extended quasi-overlap functions

Overlap functions, as a class of not necessarily associative novel aggregation functions, have played an important role in the relevant theory and applications involving fuzzy sets and systems. Recently, extension study has become a hot topic in the theoretical research of overlap functions, especially for the so-called quasi-overlap functions. In this paper, we investigate the set-based extended quasi-overlap functions, which are one kind of set -based extended aggregation functions proposed by Mesiar et al. lately. Specifically, after introducing the definition of set-based extended quasi-overlap functions, we study some of their elementary properties. Furthermore, we give two equivalent characterizations of set-based extended quasi-overlap functions by means of mappings from the set composed by all finite subsets of unit closed interval with no repetition of elements to unit closed interval and 0,1-aggregation functions recently proposed by the author, respectively. In particular, it is shown that the family of set-based extended quasi-overlap functions is closed to convex combination and that the repetition of values in the input vectors to be aggregated and adding any value between its minimum and maximum values have no effect on the output value under the aggregation of set-based extended quasi-overlap func-tions. The obtained results afford theoretical basis for more potential applications of quasi -overlap functions in real problems, especially when there are more than two classes or objects in fuzzy context or fuzzy sets for which we need to measure the degree of overlap-ping or generate overlap indices.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

Overlap functions are a class of aggregation functions that have been widely used in the theory and applications of fuzzy sets and systems. The study of extension and properties of overlap functions, especially quasi-overlap functions, has become a hot topic recently. In this paper, the authors investigate the set-based extended quasi-overlap functions and study their properties. The results show that the family of set-based extended quasi-overlap functions is closed to convex combination and is not affected by repetition of values or the addition of values between the minimum and maximum values in the input vectors.