Elsevier

Fuzzy Sets and Systems

Volume 357, 15 February 2019, Pages 58-90
Fuzzy Sets and Systems

On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and grouping functions

https://doi.org/10.1016/j.fss.2018.06.001Get rights and content

Abstract

In this paper, firstly, we introduce the notions of pseudo-homogeneous overlap and grouping functions, which can be regarded as the generalizations of the concepts of homogeneous and quasi-homogenous overlap and grouping functions, respectively. And then, we investigate the homogeneity, quasi-homogeneity and pseudo-homogeneity of an overlap function when it belongs to one of the classes of idempotent overlap functions, multiplicatively generated overlap functions or the ordinal sum of overlap functions. Finally, we show an analogous discussion for grouping functions.

Section snippets

A brief introduction to overlap and grouping functions

Recently, as two peculiar cases of binary aggregation functions [7], [32], Bustince et al. introduced the concepts of overlap and grouping functions, respectively, in [10] and in [12]. These two concepts originate from some problems in image processing and classification. In image processing, see, e.g., in [9], Bustince et al. used the so-called restricted equivalence functions to compute the threshold of an image. In classification, see, e.g., in [2], Amo et al. discussed the classification

Preliminaries

In this section, we recall some fundamental concepts and results which shall be needed in the sequel. At first, we introduce the concepts of overlap and grouping functions. Readers can refer to [6], [11], [12], [18], [19], [22], [35] for more properties of overlap and grouping functions, such as migrativity, homogeneity, idempotency, convex combination, Archimedean and so on.

Definition 2.1

(See Bustince et al. [10].) A binary function O:[0,1]2[0,1] is said to be an overlap function if, for all x,y[0,1], it

Some new results for multiplicatively generated overlap and grouping functions

In this section, we show several new results for multiplicatively generated overlap and grouping functions, which shall be used in Sections 4 and 5. We begin with the situation for overlap functions.

Proposition 3.1

Let O:[0,1]2[0,1] be an overlap function multiplicatively generated by the pair (g,h). Then g(1)=1.

Proof

Since g is an increasing function and h(1)21, one has thatg(1)g(h(1)2)=O(1,1)=1by condition (O3) of Definition 2.1. Thus, we get that g(1)=1. 

Proposition 3.2

Let O:[0,1]2[0,1] be an overlap function

Homogeneous, quasi-homogeneous and pseudo-homogeneous overlap functions

In this section, at first, we introduce the definition of pseudo-homogeneous overlap functions by a binary function F and propose some vital properties of F. And then, we discuss the pseudo-homogeneity of an overlap function when it is idempotent, multiplicatively generated or as an ordinal sum of overlap functions, respectively. In particular, we investigate the homogeneity and quasi-homogeneity of an overlap function when it belongs to one of the preceding three classes.

Definition 4.1

An overlap function O:

Homogeneous, quasi-homogeneous and pseudo-homogeneous grouping functions

In this section, firstly, we give the concept of pseudo-homogeneous grouping functions by a binary function F and show some vital properties of F. And then, we investigate the pseudo-homogeneity of a grouping function when it is idempotent, multiplicatively generated or as an ordinal sum of grouping functions, respectively. As a consequence, we propose the homogeneity and quasi-homogeneity of a grouping function when it belongs to one of the preceding three classes. It should be pointed out

Concluding remarks

This paper proposes the topics of homogeneity, quasi-homogeneity and pseudo-homogeneity for two kinds of special aggregation functions. We focus on these three vital properties for overlap and grouping functions. The main conclusions are listed as follows.

  • (1)

    In order to discuss the homogeneity, quasi-homogeneity and pseudo-homogeneity of overlap and grouping functions, we give a characterization to multiplicatively generator pairs of multiplicatively generated overlap functions and

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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