New results on the modularity condition for overlap and grouping functions

doi:10.1016/j.fss.2019.10.014

Fuzzy Sets and Systems

Volume 403, 15 January 2021, Pages 139-147

https://doi.org/10.1016/j.fss.2019.10.014 Get rights and content

Abstract

Recently, Wang and Liu investigated the modularity condition for overlap and grouping functions [Fuzzy Sets Syst. 372 (2019) 97-110]. They characterized the modularity equation between overlap (grouping) functions and proper uninorms, which belong to the most studied classes. In this paper, as a supplement of the above paper, we explore some new results on the modularity condition for overlap (grouping) functions and uninorms.

Section snippets

A brief review of overlap and grouping functions

In recently years, as two kinds of not necessarily associative fuzzy logic connectives, overlap functions [4] and grouping functions [5] were introduced by Bustince et al. in 2009 and 2012, respectively. Those two concepts originated from some problems in image processing, classification and decision making. In addition, in the past few years, overlap and grouping functions have been a rapid development both in theory and applications.

In theory, there exist many discussions involving various

Preliminaries

In this section, we briefly introduce some basic concepts and definitions related to t-norms, t-conorms, overlap (grouping) functions and uninorms, which shall be needed in the sequel.

Definition 2.1

[25] 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: $T (x, 1) = x$ .

Definition 2.2

[25] A bivariate function $S : [0,$

The new results

In this section, we explore some new results on the modularity condition for overlap functions over uninorms when the uninorms are disjunctive or conjunctive and locally internal on the boundary, and the modularity condition for uninorms over grouping functions when the uninorms are conjunctive or disjunctive and locally internal on the boundary. We presuppose that neutral element $e \in (0, 1)$ because the modularity condition between overlap (grouping) functions and t-norms (t-conorms) has been

Conclusions

In this paper, we explore some new results on the modularity condition for overlap (grouping) functions and uninorms. This work is a further investigation on the topic of modularity condition among specific binary aggregation functions and we mainly investigate Eq. (1) when the uninorms are disjunctive or conjunctive and locally internal on the boundary, and Eq. (2) when the uninorms are conjunctive or disjunctive and locally internal on the boundary. The results we obtained are complete 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. 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 (46)

B. Bedregal et al.
Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions
Int. J. Approx. Reason.
(2017)
B. Bedregal et al.
New results on overlap and grouping functions
Inf. Sci.
(2013)
T. Calvo
On some solutions of distributivity equation
Fuzzy Sets Syst.
(1999)
T. Calvo et al.
The functional equations of Frank and Alsina for uninorms and nullnorms
Fuzzy Sets Syst.
(2001)
G.P. Dimuro et al.
Generalized $C_{F_{1} F_{2}}$ -integrals: from Choquet-like aggregation to ordered directionally monotone functions
Fuzzy Sets Syst.
(2020)
G.P. Dimuro et al.
Archimedean overlap functions: the ordinal sum and the cancellation, idempotency and limiting properties
Fuzzy Sets Syst.
(2014)
G.P. Dimuro et al.
On ( $G, N$ )-implications derived from grouping functions
Inf. Sci.
(2014)
M. Elkano et al.
Fuzzy rule-based classification systems for multi-class problems using binary decomposition strategies: on the inflence of n-dimensional overlap functions in the fuzzy reasoning method
Inf. Sci.
(2016)
W. Fechner et al.
The modularity on some classes of aggregation operators
Fuzzy Sets Syst.
(2018)
D. Gómez et al.
n-dimensional overlap functions
Fuzzy Sets Syst.
(2016)

D. Jočić et al.

Restricted distributivity for aggregation operators with absorbing element

Fuzzy Sets Syst.

(2013)

A. Jurio et al.

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

Fuzzy Sets Syst.

(2013)

Y. Li et al.

Remarks on uninorms aggregation operators

Fuzzy Sets Syst.

(2000)

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)

G. Lucca et al.

$C_{F}$ -integrals: a new family of pre-aggregation functions with application to fuzzy rule-based classification systems

Inf. Sci.

(2018)

M. Mas et al.

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

Fuzzy Sets Syst.

(2015)

M. Mas et al.

The modularity condition for uninorms and t-operators

Fuzzy Sets Syst.

(2002)

M. Mas et al.

The distributivity condistions for uninorms and t-operators

Fuzzy Sets Syst.

(2002)

L. De Miguel et al.

Gerneral overlap functions

Fuzzy Sets Syst.

(2019)

J. Qiao et al.

On generalized migrativity for overlap functions

Fuzzy Sets Syst.

(2019)

J. Qiao et al.

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

Fuzzy Sets Syst.

(2019)

J. Qiao et al.

On multiplicative generators of overlap and grouping functions

Fuzzy Sets Syst.

(2018)

Y. Su et al.

On the distributivity property for uninorms

Fuzzy Sets Syst.

(2016)

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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.
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The known results indicated that this feature plays a very significant role in lots of practical application fields such as decision making [16], classification [1,15,30,32], image processing [6,25], fuzzy community problem [22] and so on. Furthermore, in terms of the theory, there exist many discussions involving various aspects of overlap functions, such as the work related to the vital properties [54–56], the ordinal sum [11], the additive and multiplicative generator pairs [12,38], the interval overlap functions [1,5,10,40], the corresponding distributive laws [29,41,42], the binary relations induced from overlap functions [43], the related concept extension studies [23,36] and so on. The notion of Archimedean overlap functions is introduced by Dimuro and Bedregal [11], which provide a research on cancellation, idempotency and limiting properties, and presented features for these functions.
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.
New extensions of quasi-overlap functions and their generalized forms on bounded posets via ⋄-operators
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As one class of binary continuous aggregation functions which play an important role in practical applications, overlap functions defined on the unit closed interval have been developed rapidly in the past decade. At the same time, Paiva et al. recently extended the concept of overlap functions on unit closed interval to the lattice-valued status and called them quasi-overlap functions on bounded lattices. In this paper, we mainly study the extension methods of quasi-overlap functions and their three generalized forms on bounded partially ordered sets. More concretely, first, we show some new extension methods of quasi-overlap functions, $0_{P}$ -quasi-overlap functions, $1_{P}$ -quasi-overlap functions and $0_{P}, 1_{P}$ -quasi-overlap functions on any bounded partially ordered set P by the so-called ⋄-operators and 0,1-homomorphisms, 1-⋄-operators and 1-homomorphisms, 0-⋄-operators and 0-homomorphisms, and 0,1-⋄-operators and ord-homomorphisms, respectively, which are different from the extension methods obtained by Qiao lately. And then, as an application of the new extension methods, some concrete quasi-overlap functions, $0_{P}$ -quasi-overlap functions, $1_{P}$ -quasi-overlap functions and $0_{P}, 1_{P}$ -quasi-overlap functions on some certain bounded partially ordered set P are constructed. Finally, we prove that these extensions maintain idempotent and Archimedean property of the known quasi-overlap functions, $0_{P}$ -quasi-overlap functions, $1_{P}$ -quasi-overlap functions and $0_{P}, 1_{P}$ -quasi-overlap functions on any bounded partially ordered set P.
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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.
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Short communicationNew results on the modularity condition for overlap and grouping functions

Abstract

Section snippets

A brief review of overlap and grouping functions

Preliminaries

The new results

Conclusions

Acknowledgements

Int. J. Approx. Reason.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Knowl.-Based Syst.

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.

Short communication
New results on the modularity condition for overlap and grouping functions