Elsevier

Fuzzy Sets and Systems

Volume 346, 1 September 2018, Pages 1-54
Fuzzy Sets and Systems

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

https://doi.org/10.1016/j.fss.2017.11.012Get rights and content

Abstract

In this paper, we investigate the migrativity of uninorms and nullnorms over any fixed overlap function or grouping function. First, we introduce the concept of (α,O)-migrative uninorms over any fixed overlap function O. In addition, we show equivalent characterizations of the (α,O)-migrativity equation when the uninorm U belongs to one certain class (e.g. Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). And then, we give the notion of (α,G)-migrative uninorms over any fixed grouping function G and propose the (α,G)-migrativity equation using an analogous method. Finally, we discuss the (α,O)-migrative and (α,G)-migrative nullnorms and obtain equivalent characterizations of the related (α,O)-migrativity and (α,G)-migrativity equations, respectively.

Section snippets

A brief review of overlap and grouping functions

Two special cases of binary aggregation functions [9], [36], overlap and grouping functions [11], [15] are introduced, respectively, by Bustince et al. in 2009 and 2012. Those two concepts originate from some practical problems in image processing [10], decision making [64] or classification [4], [29]. Recently, overlap and grouping functions have seen a rapid development both in application and theory.

  • In the applications, in 2010, Bustince et al. [12] proposed an object recognition problem,

Preliminaries

In this section, we recall some fundamental concepts and definitions which shall be needed in the sequel. At first, we give a brief reminder of the definitions of t-norms and t-conorms.

Definition 2.1

See Klement et al. [41]

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))=T(T(x,y),z);

  • (T3)

    Monotonicity: T(x,y)T(x,z) whenever yz;

  • (T4)

    Boundary condition: T(x,1)=x.

    Moreover, a t-norm T is said to be

  • (T5)

    continuous if it is

Migrativity property of uninorms over overlap functions

In this section, firstly, we introduce the concept of the migrativity property of a uninorm U over an overlap function O. Then, we investigate the related vital properties for migrativity equations of uninorms over a fixed overlap function. In particular, we discuss the situations when the uninorm becomes a t-norm or a t-conorm. We begin with the definition of the migrativity property of uninorms over a given overlap function.

Definition 3.1

Consider α in [0,1] and a given overlap function O. A uninorm U:[0,1]2

Migrativity property for some usual classes of uninorms over overlap functions

In this section, we propose the (α,O)-migrative uninorm U when U belongs to one of the usual classes, that is, UUmin,Umax,Urep,Uidor Ucos. In addition, since we have discussed (α,O)-migrative uninorm when e{0,1}, it follows that, in the sequel, we only consider the situation for e]0,1[.

Migrativity property of uninorms over grouping functions

In this section, at first, we introduce the definition of the migrativity property of a uninorm U over a grouping function G. Then, we discuss the migrativity property of the usual classes of uninorms over any fixed grouping function. We begin with the definition of migrativity property of uninorms over a given grouping function.

Definition 5.1

Consider α in [0,1] and a given grouping function G. A uninorm U:[0,1]2[0,1] is said to be α-migrative over G ((α,G)-migrative, for short) ifU(G(α,x),y)=U(x,G(α,y))

Migrativity property of nullnorms over overlap and grouping functions

In this section, firstly, we introduce the concept of the migrativity property of a nullnorm F over an overlap function O and give characterizations of the migrativity property of nullnorms over any fixed overlap function then we investigate the migrativity property of nullnorms over any fixed grouping function in a similar way.

Concluding remarks

This paper investigates the topic of α-migrativity of one class of special aggregation functions over any fixed special aggregation function. We introduce the notions of α-migrativity for uninorms and nullnorms by taking the new class of special aggregation functions, overlap and grouping functions, as the fixed special aggregation functions in the α-migrativity equations, respectively. And those α-migrativity equations are proposed and characterized on the different situations. The main

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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