A short note on the migrativity properties of overlap functions over uninorms

doi:10.1016/j.fss.2020.06.011

Fuzzy Sets and Systems

Volume 414, 1 July 2021, Pages 135-145

https://doi.org/10.1016/j.fss.2020.06.011 Get rights and content

Abstract

Recently, as an addendum to Qiao and Hu (2018) [21], Zhu and Hu (2020) [25] studied the α-migrativity of overlap and grouping functions over uninorms and nullnorms. Later on, Zhou and Yan (2019) [24] further investigated the α-migrativity properties of overlap functions over uninorms. They showed equivalent characterizations of the $(α, U)$ -migrativity equation when the uninorm U belongs to one certain class (e.g., $U_{\min}$ , $U_{\max}$ , the family of idempotent uninorms, representable uninorms or uninorms continuous on $] 0, 1 [^{2}$ ). However, we find that Propositions 3.5, 3.8 and Remark 3.5 (2) presented by Zhu and Hu (2020) [25] are incorrect. In this note, at first, we establish the updated results. And then, we study the α-migrativity of overlap functions over t-norms when t-norms are continuous and give their characterizations. In particular, we point out that the relationship between the α-migrativity of overlap functions over uninorms and the α-migrativity of uninorms over overlap functions. Finally, we discuss the $(α, U)$ -migrativity equation for grouping functions using an analogous method.

Section snippets

Introduction and motivation

The α-migrativity as an interesting property of two-place functions on [0,1] has been proposed in many works. There are many researches which have pointed out that the investigation of α-migrativity for aggregation functions or some special binary aggregation functions has a vital meaning and value both in applications and theory (about the introduction of α-migrativity, see Introduction in [24]). In particular, it should be pointed out that Bedregal et al. [4] first studied on α-migrativity of

Preliminaries

In this section, we recall some basic 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

[16] A bivariate function $T : {[0, 1]}^{2} ⟶ [0, 1]$ is said to be a t-norm if, for all $x, y, z \in [0, 1]$ , it satisfies the following conditions:

(T1) Commutativity: $T (x, y) = T (y, x)$ ;

(T2) Associativity: $T (T (x, y), z) = T (x, T (y, z))$ ;

(T3) Monotonicity: $T (x, y) \leq T (x, z)$ whenever $y \leq z$ ;

(T4) Boundary condition: $T (x, 1) = x$ .

Moreover, a t-norm T is said to be

Discussion about some incorrect results presented in [25]

In this section, at first, we point out that some incorrect results presented in [25]. Then, we provide a new version of such results, correcting all the inconsistencies and also give the proof.

New results

In this section, we explore some new results on the α-migrativity of overlap functions over continuous t-norms and uninorms with neutral element $e \in] 0, 1 [$ , and we do an analogous study for the α-migrativity of grouping functions over continuous t-conorms and uninorms with neutral element $e \in] 0, 1 [$ .

At first, it follows from Proposition 3.1 that for any $α \in] 0, 1 [$ , an overlap function O is not α-migrative over a t-conorm S. Thus, we only consider the α-migrativity for overlap functions over t-norms and

Conclusions

In this note, at first, we point out that the statements of Propositions 3.5, 3.8 and Remark 3.5 (2) in [25] are incorrect and provide their modified versions. Then, we explore some new results on the α-migrativity between overlap (grouping) functions and uninorms. This work is a further investigation on the topic of α-migrativity among specific binary aggregation functions and we mainly investigate the α-migrativity of overlap functions over t-norms when t-norms are continuous. In particular,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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. This work was supported in part by Higher Education Key Scientific Research Program Funded by Henan Province (No. 20A110011) and Research and Cultivation Fund Project of Anyang Normal University (No. AYNUKP-2018-B26).

References (25)

B. Bedregal et al.
New results on overlap and grouping functions
Inf. Sci.
(2013)
H. Bustince et al.
Overlap functions
Nonlinear Anal.
(2010)
H. Bustince et al.
A generalization of the migrativity property of aggregation functions
Inf. Sci.
(2012)
T. Calvo et al.
The functional equations of Frank and Alsina for uninorms and nullnorms
Fuzzy Sets Syst.
(2001)
G.P. Dimuro et al.
Archimedean overlap functions: the ordinal sum and the cancellation, idempotency and limiting properties
Fuzzy Sets Syst.
(2014)
F. Durante et al.
A note on the convex combinations of triangular norms
Fuzzy Sets Syst.
(2008)
J. Fodor et al.
An extension of the migrative property for triangular norms
Fuzzy Sets Syst.
(2011)
S.K. Hu et al.
The structure of continuous uninorms
Fuzzy Sets Syst.
(2001)
A. Jurio et al.
Some properties of overlap and grouping functions and their application to image thresholding
Fuzzy Sets Syst.
(2013)
M. Mas et al.
Migrative uninorms and nullnorms over t-norms and t-conorms
Fuzzy Sets Syst.
(2015)

J. Qiao et al.

On the migrativity of uninorms and nullnorms over 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)

Cited by (11)

On the migrativity of 2-uninorms
2023, Fuzzy Sets and Systems
This article focuses on the α-migrativity of 2-uninorms with $α \in [0, 1]$ . It characterizes the α-migrativity for 2-uninorms with $α \in [0, 1]$ , which generalizes and unifies the characterizations for some existing α-migrativity of uni-nullnorms, null-uninorms, uninorms and nullnorms, respectively. It also proves that for every 2-uninorm $U_{1}$ (resp. $U_{2}$ ) and $α \in [0, 1]$ , there always exists a 2-uninorm $U_{2} \neq U_{1}$ (resp., $U_{1} \neq U_{2}$ ) such that $U_{1}$ is α-migrative over $U_{2}$ .
Characterizations for the cross-migrativity between overlap functions and commutative aggregation functions
2023, Information Sciences
Previous years, people deeply discussed the cross-migrativity properties of the same binary operators, like two overlap functions. Based on the previous works, we extend the study of the cross-migrativity about two same operators to different operators (i.e., a commutative aggregation function $C_{A}$ and overlap function F). At first, we present a idea respecting cross-migrativity about commutative aggregation functions with overlap operators. Besides, we obtain some equivalent characterizations of $C_{A}$ is cross-migrative over $F_{M}$ and $F_{w}$ , respectively. Moreover, by the aid of ordinal sum about $C_{A}$ , some related properties on the cross-migrativity equations between $C_{A}$ and $F_{M}$ and $F_{1}$ are discussed, respectively.
Characterizations for the migrativity of continuous t-conorms over fuzzy implications
2023, Fuzzy Sets and Systems
Citation Excerpt :
Because of its wide applications, especially in image processing [27] and decision making [6,31,32], the migrativity property has aroused great interest of researchers. The migrativity of semi-copulas, copulas [10,27], uninorms, nullnorms [1,17,22,24,25,39–42,60], triangular subnorms, triangular norms (t-norms, for short) [13–15,33,46], and overlap functions [36–38,56,57,59] have been extensively studied with fruitful results. The main results of this paper are characterizations of solutions to some α-migrativity functional equations for t-conorms over fuzzy implications, which are summarized in Table 1 at the end of the paper.
Migrativity between fuzzy logical connectives is an important property at both sides of theory and practical application in fuzzy logic. The migrativity properties between conjunctive connectives as well as general aggregation functions including t-norms, uninorms, nullnorms and overlap functions have been extensively investigated. Recently, Baczyński et al. (2020) [3] studied the migrativity of fuzzy implications. However, there are no discussions so far on the migrativity between disjunctive connectives and fuzzy implications. The purpose of the present paper is to characterize the migrativity of continuous t-conorms over fuzzy implications, which also provides an answer to Fodor's question on how to define the migrativity of t-conorms. We first describe completely the α-migrativity of continuous t-conorms defined by Baczyński et al. in their latest work, which proves to be the standard dual to the migrativity of t-norms over the product t-norm, but provides a new perspective to study the migrativity of t-conorms. And then, we turn to characterize the migrativity of t-conorms over several specific well-known fuzzy implications, of which some are no longer dual to the migrativity of t-norms, and show some interesting results. Finally, we define α-migrativity of continuous t-conorms over general fuzzy implications and obtain the characterizations of solutions to migrativity equations by the ordinal sum of t-conorms, where some supporting examples for solutions are given.
On the cross-migrativity between uninorms and overlap (grouping) functions
2022, Fuzzy Sets and Systems
Overlap (grouping) functions, uninorms and nullnorms, as sorts of aggregation functions, have been widely applied in decision making, classification, image processing and other related fields. On the other hand, α-migrativity as a crucial and particularly interesting feature of bivariate aggregation functions has been researched in many literatures. In this paper, firstly, based on the α-cross-migrativity between an overlap function O and a t-norm T, we clarify the α-cross-migrativity between O and a uninorm U when α takes some special values. Afterwards, we confirm that disjunctive uninorms and overlap functions are not α-cross-migrative. Then, we prove α-cross-migrativity equation when the uninorm U belongs to five general classes, i.e., $U_{\min}$ , $U_{\max}$ , $U_{i d e}$ , $U_{r e p}$ and $U_{\cos, \min}$ in detail. In a similar way, the α-cross-migrativity between a grouping function G and a uninorm U is discussed by duality. Finally, we consider the α-cross-migrativity between nullnorms and overlap (grouping) functions.
On ordinal sums of overlap and grouping functions on complete lattices
2022, Fuzzy Sets and Systems
Citation Excerpt :
As fuzzy logical operators that are not necessarily associative, overlap and grouping functions play an expressive part in applications, such as image processing [4,30], decision making [5,22], classification [17,18,20,21,32–35,44], fuzzy community detection problems [28] and other fields [26,40]. Theoretically, key attributes such as migrativity, homogeneity, idempotency, cancellation law, modularity and relative distributive equations of overlap and grouping functions on the unit interval have been studied in [2,14,45,48–50,58–64]. Also, researchers considered construction methods of overlap and grouping functions, including ordinal sums [14,50], additive generators [15,16] and multiplicative generators [47] on the unit interval.
Recently, Qiao gave the concrete form of the continuity of overlap and grouping functions on complete lattices by using meet- and join-preserving properties. By eliminating the property join-preserving (resp. meet-preserving), one gets right (resp. left) continuous functions, namely, $C_{R}$ - (resp. $C_{L}$ -) overlap and grouping functions. In this paper, we extend the notion of ordinal sums of overlap functions from the unit interval to complete lattices directly and indicate it does not necessarily lead to an overlap function. We find that the ordinal sums of finitely many overlap functions can create an overlap function on a frame that can be partitioned into a chain of subintervals. We also investigate ordinal sums of $C_{R}$ - and $C_{L}$ -overlap functions on complete lattices, where the endpoints of summand carriers constitute a chain. Moreover, we have an analogous discussion on grouping functions on complete lattices.
Some new results on the migrativity of uninorms over overlap and grouping functions
2022, Fuzzy Sets and Systems
In 2018, Qiao and Hu [48] studied the α-migrativity of uninorms over overlap and grouping functions when the uninorm U belongs to one certain class (e.g., $U_{\min}$ , $U_{\max}$ , the family of idempotent uninorms, representable uninorms or uninorms continuous on $] 0, 1 [^{2}$ ). In addition, they obtained some equivalent characterizations of them. This paper will continue to consider the characterizations of this kind of migrativity equations by means of the ordinal sum of overlap and grouping functions. We give the necessary and sufficient conditions for the solutions of the $(α, O)$ -migrativity equation when the uninorm U becomes a t-norm or a conjunctive uninorm locally internal on the boundary and the $(α, G)$ -migrativity equation when the uninorm U becomes a t-conorm or a disjunctive uninorm locally internal on the boundary, respectively.

View all citing articles on Scopus

View full text

Short communicationA short note on the migrativity properties of overlap functions over uninorms

Abstract

Section snippets

Introduction and motivation

Preliminaries

Discussion about some incorrect results presented in [25]

New results

Conclusions

Declaration of Competing Interest

Acknowledgements

Inf. Sci.

Nonlinear Anal.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Short communication
A short note on the migrativity properties of overlap functions over uninorms