On generalized migrativity property for overlap functions
Section snippets
A brief review on overlap functions
The concept of overlap functions [11] has been given by Bustince et al. in 2009. It arises from some problems in image processing and classification. For instance, in image processing, see, e.g., Bustince et al. [10] used the so-called restricted equivalence functions for the computation of the threshold of an image. In classification, see, e.g., Amo et al. [6] investigated the classification system [53] when the classes of objects are fuzzy in practice. Thus, the overlapping is used to study
Preliminaries
In this section, we recall some fundamental concepts and definitions which shall be needed in the sequel. We begin with the definition of t-norms.
Notice that the condition of increase/decrease for the functions involved in this paper always denotes the weak increase/decrease. Definition 2.1 (See Alsina et al. [3], Klement et al. [35].) A bivariate function is said to be a t-norm if, for all , it satisfies the following conditions: Commutativity: ; Associativity:
Generalizing the migrativity property by overlap functions
In this section, firstly, we extend the α-migrativity of any overlap function O to the so-called -migrativity on the basis of two fixed overlap functions and . Secondly, based on the ordinal sum of overlap functions, we show an equivalent characterization of the -migrativity of an overlap function by considering and as the minimum overlap function. Thirdly, we give an equivalent characterization of the -migrativity of an overlap function through taking
Concluding remarks
In this paper, we introduced the concept of -migrativity of overlap functions, where and are two fixed overlap functions. We also characterized three special classes of -migrativity of overlap functions. The main conclusions are listed as follows.
- (1)
We generalized the α-migrativity of any overlap function O from the usual formula to the so-called -migrativity , where and are two fixed overlap functions.
- (2)
We discussed the
Acknowledgements
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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