Discrete overlap functions: Basic properties and constructions

Abstract

As a kind of emerging binary continuous aggregation operator that has been successfully applied in many practical application problems, overlap functions on the unit closed interval have been considered by scholars on different truth values sets lately. At the same time, studying aggregation operators on finite chains, especially for commonly used binary aggregation operators, is a meaningful and hot topic in the research field of aggregation operators. In this paper, we pay attention to overlap functions on finite chains, which are called discrete overlap functions. Specifically, first, we introduce the notions of discrete overlap functions on the finite chain L with n+2 elements and its arbitrary subchains along with an extended form of them. Second, we study some basic properties of discrete overlap functions on L, especially for the idempotent property, Archimedean property and cancellation law. In particular, we obtain some new properties which are different from those of the overlap functions on other truth values sets, for instance, every discrete overlap function on L takes the greatest element on L as the neutral element. Third, we discuss the construction methods of discrete overlap functions on L. Finally, it is worth mentioning that the results obtained in this paper provide a theoretical basis and more possibilities for the potential applications of overlap functions in other fields besides their known applications, especially for the situation of that the reasoning of experts are described by linguistic terms or labels, such as in expert systems, fuzzy control and etc.