On generalized migrativity property for overlap functions

Abstract

Overlap functions, as a new special class of binary aggregation functions, have been investigated in many recent works for applications in image processing, classification problems and decision making. In addition, α-migrativity as a vital property of binary functions on unit interval has been discussed in the literature. In particular, recently, an interesting and natural research topic for α-migrativity is the investigation of the generalized forms of α-migrativity for one peculiar binary aggregation functions over any fixed special binary aggregation function (see, e.g., the α-migrativity of t-norms over any fixed t-norm, t-conorms over any fixed t-conorm, uninorms over any fixed uninorm, and uninorms and nullnorms, respectively, over any fixed t-norm or t-conorm.). Thus, in this paper, we continue to study this research topic for overlap functions. At first, we generalize the α-migrativity of any overlap function O from the usual formula O(αx,y)=O(x,αy) to the so-called (α,O⁎,O†)-migrativity O(O⁎(α,x),y)=O(x,O†(α,y)), where O⁎ and O† are two fixed overlap functions. And then, we investigate the (α,O⁎,O†)-migrativity of an overlap function by taking O⁎ and O† as the minimum overlap function and give an equivalent characterization of it by the ordinal sum of overlap functions. In addition, we propose the (α,O⁎,O†)-migrativity of an overlap function by taking O⁎ and O† as the p-product overlap function and show an equivalent characterization of it by its additive generator pair. In particular, we obtain an equivalent characterization of the usual α-migrativity of an overlap function by its additive generator pair. Finally, we discuss the (α,O⁎,O†)-migrativity of an overlap function by taking O⁎ and O† as the 1-product overlap function and p-product overlap function, respectively, and give two characterizations of it by its additive generator pair.