On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and 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 is said to be an overlap function if, for all , 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 be an overlap function multiplicatively generated by the pair . Then . Proof Since g is an increasing function and , one has that Thus, we get that . □ Proposition 3.2 Let 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
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).
References (56)
- et al.
Fuzzy classification systems
Eur. J. Oper. Res.
(2004) - et al.
Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions
Int. J. Approx. Reason.
(2017) - et al.
New results on overlap and grouping functions
Inf. Sci.
(2013) - et al.
Contrast of a fuzzy relation
Inf. Sci.
(2010) - et al.
Image thresholding using restricted equivalent functions and maximizing the measures of similarity
Fuzzy Sets Syst.
(2007) - et al.
Overlap functions
Nonlinear Anal.
(2010) Solutions to homogeneous Monge–Ampère equations of homothetic functions and their applications to production models in economics
J. Math. Anal. Appl.
(2014)- et al.
Archimedean overlap functions: the ordinal sum and the cancellation, idempotency and limiting properties
Fuzzy Sets Syst.
(2014) - et al.
QL-operations and QL-implication functions constructed from tuples and the generation of fuzzy subsethood and entropy measures
Int. J. Approx. Reason.
(2017) - et al.
On -implications derived from grouping functions
Inf. Sci.
(2014)
Fuzzy Rule-Based Classification Systems for multi-class problems using binary decomposition strategies: on the influence of n-dimensional overlap functions in the Fuzzy Reasoning Method
Inf. Sci.
Consensus via penalty functions for decision making in ensembles in fuzzy rule-based classification systems
Appl. Soft Comput.
On -homogeneous copulas
Inf. Sci.
n-dimensional overlap functions
Fuzzy Sets Syst.
Some properties of overlap and grouping functions and their application to image thresholding
Fuzzy Sets Syst.
An interval extension of homogeneous and pseudo-homogeneous t-norms and t-conorms
Inf. Sci.
On fuzzy complements
Inf. Sci.
CC-integrals: Choquet-like Copula-based aggregation functions and its application in fuzzy rule-based classification systems
Knowl.-Based Syst.
On interval additive generators of interval overlap functions and interval grouping functions
Fuzzy Sets Syst.
On the migrativity of uninorms and nullnorms over overlap and grouping functions
Fuzzy Sets Syst.
On multiplicative generators of overlap and grouping functions
Fuzzy Sets Syst.
The distributive laws of fuzzy implications over overlap and grouping functions
Inf. Sci.
Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning
Inf. Sci.
On pseudo-homogeneous triangular norms, triangular conorms and proper uninorms
Fuzzy Sets Syst.
Fuzzy sets
Inf. Control
Associative Functions: Triangular Norms and Copulas
On the distributivity of fuzzy implications over nilpotent or strict triangular conorms
IEEE Trans. Fuzzy Syst.
Using the Choquet integral in the fuzzy reasoning method of fuzzy rule-based classification systems
Axioms
Cited by (41)
Note on the pseudo-homogeneous overlap and grouping functions
2023, Fuzzy Sets and SystemsOn fuzzy Sheffer strokes: New results and the ordinal sums
2023, Fuzzy Sets and SystemsNote on the homogeneity of overlap functions
2023, Fuzzy Sets and SystemsCitation Excerpt :As a particular case of bivariate continuous aggregation functions, overlap functions were introduced by Bustince et al. [3,4], aiming at solving problems arisen from image processing, classification or decision making for which t-norms are usually considered but where associativity is not needed. Recently, the overlap functions have a rapid development both in theory and applications (see, e.g., [2,6,7,9–11,14]). As it is well-known, a very important way for building aggregation operations is by additive generators (see, e.g., [8,12,13]).
Pre-(quasi-)overlap functions on bounded posets
2022, Fuzzy Sets and SystemsConstructing general overlap and grouping functions via multiplicative generators
2022, Fuzzy Sets and SystemsCitation Excerpt :The notion dual of overlap functions were called of grouping functions in [5]. From then on, grouping functions have been extensively studied as for example in [24–26], used to generate fuzzy implications in [8,12,13] and applied in [17–19]. Naturally, generalizations duals to the done for overlap functions also were investigated, given arising to the notions of n-dimensional grouping functions [17] and general grouping functions [27].