Overlap functions

In this paper we address a key issue in scenario classification, where classifying concepts show a natural overlapping. In fact, overlapping needs to be evaluated whenever classes are not crisp, in order to be able to check if a certain classification structure fits reality and still can be useful for our declared decision making purposes. In this paper we address an object recognition problem, where the best classification with respect to background is the one with less overlapping between the class object and the class background. In particular, in this paper we present the basic properties that must be fulfilled by overlap functions, associated to the degree of overlapping between two classes. In order to define these overlap functions we take as reference properties like migrativity, homogeneity of order 1 and homogeneity of order 2. Hence we define overlap functions, proposing a construction method and analyzing the conditions ensuring that t-norms are overlap functions. In addition, we present a characterization of migrative and strict overlap functions by means of automorphisms.