Elsevier

Fuzzy Sets and Systems

Volume 357, 15 February 2019, Pages 91-116
Fuzzy Sets and Systems

On generalized migrativity property for overlap functions

https://doi.org/10.1016/j.fss.2018.01.007Get rights and content

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.

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 T:[0,1]2[0,1] is said to be a t-norm if, for all x,y,z[0,1], it satisfies the following conditions:

  • (T1)

    Commutativity: T(x,y)=T(y,x);

  • (T2)

    Associativity: T(x,T(y,z))

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 (α,O,O)-migrativity on the basis of two fixed overlap functions O and O. Secondly, based on the ordinal sum of overlap functions, we show an equivalent characterization of the (α,O,O)-migrativity of an overlap function by considering O and O as the minimum overlap function. Thirdly, we give an equivalent characterization of the (α,O,O)-migrativity of an overlap function through taking O

Concluding remarks

In this paper, we introduced the concept of (α,O,O)-migrativity of overlap functions, where O and O are two fixed overlap functions. We also characterized three special classes of (α,O,O)-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 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.

  • (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).

References (59)

  • G.P. Dimuro et al.

    QL-operations and QL-implication functions constructed from tuples (O,G,N) and the generation of fuzzy subsethood and entropy measures

    Int. J. Approx. Reason.

    (2017)
  • F. Durante et al.

    On the α-migrativity of multivariate semi-copulas

    Inf. Sci.

    (2012)
  • F. Durante et al.

    A note on the convex combinations of triangular norms

    Fuzzy Sets Syst.

    (2008)
  • M. Elkano et al.

    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.

    (2016)
  • J. Fodor et al.

    On continuous triangular norms that are migrative

    Fuzzy Sets Syst.

    (2007)
  • J. Fodor et al.

    Migrative t-norms with respect to continuous ordinal sums

    Inf. Sci.

    (2011)
  • J. Fodor et al.

    An extension of the migrative property for triangular norms

    Fuzzy Sets Syst.

    (2011)
  • D. Gómez et al.

    n-Dimensional overlap functions

    Fuzzy Sets Syst.

    (2016)
  • A. Jurio et al.

    Some properties of overlap and grouping functions and their application to image thresholding

    Fuzzy Sets Syst.

    (2013)
  • C. Lopez-Molina et al.

    Bimigrativity of binary aggregation functions

    Inf. Sci.

    (2014)
  • G. Lucca et al.

    CC-integrals: Choquet-like Copula-based aggregation functions and its application in fuzzy rule-based classification systems

    Knowl.-Based Syst.

    (2017)
  • M. Mas et al.

    An extension of the migrative property for uninorms

    Inf. Sci.

    (2013)
  • M. Mas et al.

    Migrative uninorms and nullnorms over t-norms and t-conorms

    Fuzzy Sets Syst.

    (2015)
  • R. Mesiar et al.

    On the α-migrativity of semicopulas, quasi-copulas, and copulas

    Inf. Sci.

    (2010)
  • Y. Ouyang

    Generalizing the migrativity of continuous t-norms

    Fuzzy Sets Syst.

    (2013)
  • J. Qiao et al.

    On multiplicative generators of overlap and grouping functions

    Fuzzy Sets Syst.

    (2018)
  • J. Qiao et al.

    On interval additive generators of interval overlap functions and interval grouping functions

    Fuzzy Sets Syst.

    (2017)
  • J. Qiao et al.

    On the migrativity of uninorms and nullnorms over overlap and grouping functions

    Fuzzy Sets Syst.

    (2018)
  • J.A. Sanz et al.

    Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning

    Inf. Sci.

    (2010)
  • Cited by (52)

    • On the migrativity of 2-uninorms

      2023, Fuzzy Sets and Systems
    View all citing articles on Scopus
    View full text