Next Article in Journal
Numerical Solution for Singular Boundary Value Problems Using a Pair of Hybrid Nyström Techniques
Next Article in Special Issue
Normed Spaces Which Are Not Mackey Groups
Previous Article in Journal
Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations
Previous Article in Special Issue
Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

On Self-Aggregations of Min-Subgroups

1
Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, 121 35 Prague, Czech Republic
2
Departamento de Física y Matemática Aplicada, University of Navarra, 31080 Pamplona, Spain
3
Instituto de Ciencia de los Datos e Inteligencia Artificial, University of Navarra, 31080 Pamplona, Spain
*
Author to whom correspondence should be addressed.
Axioms 2021, 10(3), 201; https://doi.org/10.3390/axioms10030201
Submission received: 17 June 2021 / Revised: 9 August 2021 / Accepted: 19 August 2021 / Published: 24 August 2021

Abstract

:
Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets.

1. Introduction

Aggregation operators have become an important research topic in the last two decades. The motivation to use such functions comes from the need to summarize different pieces of information into a single object, which is a particularly challenging task when the incoming information is heterogeneous, imprecise, or incomplete. These operators are nowadays a fundamental tool of computer sciences with applications in classification, databases, control, decision making, or image processing among others. Recent monographs on this topic are [1,2,3].
An aggregation operator is a non-decreasing function A : [ 0 , 1 ] n [ 0 , 1 ] satisfying certain boundary conditions (see Definition 1). This construction allows one to aggregate not only numerical values but also any functions, or structures on a set that have output in the unit interval.
Min-subgroups were introduced by Rosenfeld in ([4]) as a fuzzy set μ whose domain is a group G such that μ ( x ) = μ ( x 1 ) and μ ( x y ) min { μ ( x ) , μ ( y ) } for all x , y in G. Note that from the definition, we immediately obtain μ ( e ) μ ( x ) for all x in G, and hence the normalization condition μ ( e ) = 1 is often added to the definition of fuzzy subgroup. Das studied min-subgroups thoroughly in [5], introducing a characterization in terms of level sets in which the level sets of μ correspond to crisp subgroups of G. Das also introduced an equivalence relation between fuzzy groups concerning level sets. Anthony and Sherwood (see [6]) extended Rosenfeld’s definition using an arbitrary t-norm T instead of the minimum. These groups are called T-subgroups. Formato and Gerla constructed a correspondence between T-indistinguishability operators on a set (relations that are reflexive, symmetric, and T-transitive) and T-subgroups related with the permutation group of the set further motivating the study of T-subgroups (see [7]).
Min-subgroups can be identified as indistinguishability operators that are invariant by translations (see [8]). This type of indistinguishability operator plays a fundamental role in some applications, notably in mathematical morphology (see [8,9,10,11]).
When the set of inputs of an aggregation function share a structure (i.e., they are all indistinguishability operators, min-subgroups, or other fuzzy relations with additional properties), the main problem is the preservation of that structure. In other words, the problem is determining conditions guarantee that the output has the same structure. Preservation of structures under aggregation has been widely studied in recent decades (see [12,13,14,15,16,17,18,19,20,21]).
In particular, preservation of the min-subgroup structure under binary aggregations was studied in [12]. However, these results cannot be immediately translated into n-ary aggregation functions since these operators are not necessarily associative. In this article, we obtain the first results concerning the preservation of the min-subgroup structure for aggregation of more than two min-subgroups. More concretely, we focus on the preservation of the min-subgroup structure under self-aggregation motivated by the central role they play in the binary case. Note that the minimum t-norm is the only t-norm that is idempotent, and it is characterized by its level-sets, which makes it very useful in certain contexts ([22]).
The remainder of the article is organized as follows. In Section 2, we introduce the relevant definitions and known facts. Section 3 contains our first new results. We show that the aggregations of an arbitrary number of min-subgroups are also min-subgroups. We also study the behavior of the fuzzy subgroup obtained from conjunctive, averaging, disjunctive, and mixed aggregation functions. Section 4 is devoted to investigate self-aggregations with respect to the equivalence classes of fuzzy subgroups given by its level sets. Our main result states that, for aggregation functions that are strictly increasing on their diagonal, the self-aggregation of a min-subgroup has the same level sets that the original min-subgroup. The article ends with some concluding remarks and future lines of research.

2. Preliminary Facts

Definition 1
([1]). An operation A : [ 0 , 1 ] n [ 0 , 1 ] is called an aggregation function if it satisfies the following axioms:
( A 1 )
Monotonicity. If x i y i for each i { 1 , , n } , then A ( x 1 , , x n ) A ( y 1 , , y n ) .
( A 2 )
Boundary conditions. A ( 0 , , 0 ) = 0 and A ( 1 , , 1 ) = 1 .
Moreover, A is called jointly strictly monotone if whenever x i < y i for all i { 1 , , n } , then A ( x 1 , , x n ) < A ( y 1 , , y n ) .
Among the most relevant aggregation functions, we find the arithmetic mean, the geometric mean, the harmonic mean, and the quadratic mean (see [1,3]). Aggregation functions are classified into four broad classes: conjunctive, averaging, disjunctive, and mixed functions.
1 .
A conjunctive aggregation function A is an aggregation function such that A ( r 1 , , r n ) min { r 1 , , r n } for all ( r 1 , , r n ) [ 0 , 1 ] n . A prototypical example is any t-norm.
2 .
An averaging aggregation function A is an aggregation function such that min { r 1 , , r n }   A ( r 1 , , r n ) max { r 1 , , r n } for all ( r 1 , , r n ) [ 0 , 1 ] n . Ordered weighted averaging operators belong to this category.
3 .
A disjunctive aggregation function A is an aggregation function such that max { r 1 , , r n }   A ( r 1 , , r n ) for all ( r 1 , , r n ) [ 0 , 1 ] n . One example is any t-conorm.
4 .
An aggregation function A is called mixed if A is not conjunctive, averaging, nor disjunctive. Uninorms belong to this type of aggregation functions.
Note that the averaging class is frequently called idempotent class since every averaging aggregation function A satisfies A ( r , , r ) = r for all ( r , , r ) [ 0 , 1 ] n . Extensive information about aggregation functions can be found in [3].
Definition 2
([4]). Let ( G , · ) be a group. We say that μ : G [ 0 , 1 ] is a min-subgroup of G if:
( G 1 )
For all x G , μ ( x ) μ ( x 1 ) .
( G 2 )
For all x , y G , μ ( x y ) min { μ ( x ) , μ ( y ) } .
Note that G 1 is equivalent to μ ( x ) = μ ( x 1 ) for all x G . In the paper, e denotes the neutral element of the group G.
Definition 3
([23]). Let μ be a fuzzy subset of a given universe X. For each t [ 0 , 1 ] , the level set μ t and strict level set μ t are defined as follows:
μ t = { x X | μ ( x ) t } μ t = { x X | μ ( x ) > t }
The support of μ is defined by supp μ = μ 0 .
Level sets (or α -cuts) have been studied extensively in fuzzy subgroups (see for instance [24,25]). P. Das used level sets to characterize the notion of min-subgroup ([5]).
Proposition 1
([5]). Let G be a group and μ a fuzzy set of G; then μ is a min-subgroup of G if and only if all its non-empty level sets are subgroups of G.

3. Self-Aggregation

Given an aggregation function A and n fuzzy subsets μ 1 , , μ n of a group G, we consider the fuzzy set A ( μ 1 , , μ n ) on G defined by
A ( μ 1 , , μ n ) ( x ) = A ( μ 1 ( x ) , , μ n ( x ) )
for each x G . We say that A ( μ 1 , , μ n ) is the aggregation of μ 1 , , μ n through A.
In this section, we will study the aggregation of A ( μ , , μ ) whenever μ is a min-subgroup of a group G, i.e., the self-aggregation of μ through A.
Anthony and Sherwood (see [6]) introduced T-subgroups as an extension of min-subgroups using an arbitrary t-norm T instead of the minimum.
The following theorem underlines the relevance of min-subgroups within T-subgroups since the minimum is the only t-norm that guarantees preservation of the T-subgroup structure for any binary self-aggregation process.
Theorem 1
([12]). Let G be a group with at least four elements and T a t-norm satisfying T T D , where T D is the drastic t-norm. The following assertions are equivalent:
1 .
T = min .
2 .
For each T-subgroup μ and each aggregation function A, A ( μ , μ ) is a T-subgroup.
Due to this result, given any aggregation function and any min-subgroup μ , A ( μ , μ ) is also a min-subgroup. However, since A is not necessarily associative, the previous result does not guarantee that A ( μ , μ , , μ ) is also a min-subgroup. We establish that this is the case for arbitrarily sized aggregations.
Proposition 2.
Let A : [ 0 , 1 ] n [ 0 , 1 ] be an aggregation function and μ a min-subgroup of a group G. Then, A ( μ , , μ ) is also a min-subgroup of G.
Proof. 
Take x G ; we have that
A ( μ , , μ ) ( x ) = A ( μ ( x ) , , μ ( x ) ) = A ( μ ( x 1 ) , , μ ( x 1 ) ) = A ( μ , , μ ) ( x 1 ) .
Take x , y G . Without loss of generality, let us assume that μ ( x ) μ ( y ) . Under this premise, using the fact that A is a non-decreasing function, we have that
A ( μ , , μ ) ( x ) = min A ( μ , , μ ) ( x ) , A ( μ , , μ ) ( y ) .
Therefore,
A ( μ , , μ ) ( x y ) = A ( μ ( x y ) , , μ ( x y ) ) A min { μ ( x ) , μ ( y ) } , , min { μ ( x ) , μ ( y ) } .
Since μ ( x ) μ ( y ) and the monotonicity of A,
A min { μ ( x ) , μ ( y ) } , , min { μ ( x ) , μ ( y ) } = A ( μ ( x ) , , μ ( x ) ) = A ( μ , , μ ) ( x ) .
Taking into account (1), the proof is completed. □
We proceed to study the comparison between A ( μ , , μ ) and μ with respect to the usual order of fuzzy sets, that is, if A ( μ , , μ ) μ or A ( μ , , μ ) μ . The following result shows sufficient conditions on A in order to compare both of them.
Proposition 3.
Let A : [ 0 , 1 ] n [ 0 , 1 ] be an aggregation function and μ a min-subgroup of a group G.
(1)
If A is a conjunctive aggregation function, then A ( μ , , μ ) μ .
(2)
If A is an averaging aggregation function, then A ( μ , , μ ) = μ .
(3)
If A is a disjunctive aggregation function, then A ( μ , , μ ) μ .
Proof. 
Let us consider x G .
(1)
A ( μ , , μ ) ( x ) = A ( μ ( x ) , , μ ( x ) ) min μ ( x ) , , μ ( x ) = μ ( x ) .
(2)
On the one hand, μ ( x ) = min μ ( x ) , , μ ( x ) A ( μ ( x ) , , μ ( x ) ) = A ( μ , , μ ) ( x ) .
On the other hand, A ( μ , , μ ) ( x ) = A ( μ ( x ) , , μ ( x ) ) max μ ( x ) , , μ ( x ) = μ ( x ) .
(3)
A ( μ , , μ ) ( x ) = A ( μ ( x ) , , μ ( x ) ) max μ ( x ) , , μ ( x ) = μ ( x ) .
However, if A is mixed, it is possible that A ( μ , , μ ) is not comparable to μ , and when it is, all the above inequalities can appear, as the following example shows.
Example 1.
Consider the group G = ( Z 6 , + ) and the fuzzy sets μ , η , ν , σ defined in the table below.
G012345
μ 0.90.50.50.90.50.5
η 10.20.80.20.80.2
ν 0.40.30.30.40.30.3
σ 1000.500
Clearly, they are min-subgroups of G because their level sets are crisp subgroups of G. Let us consider the following binary aggregation function A, where e = 0.5 is the neutral element: A ( x , y ) = y if x = e , x if y = e , 0 if x < e , y < e , 1 if x > e , y > e , e otherwise .
It is easy to check that A is a mixed aggregation function. The self-aggregations of the previous min-subgroups are:
G012345
A ( μ , μ ) 10.50.510.50.5
A ( η , η ) 101010
A ( ν , ν ) 000000
A ( σ , σ ) 1000.500
We can conclude that
A ( μ , μ ) μ ,
A ( ν , ν ) ν ,
A ( σ , σ ) = σ ,
but A ( η , η ) is not comparable to η.

4. Self-Aggregation on the Equivalence Class

There are infinitely many min-subgroups that generate the same chain of subgroups. In order attempt any classification, it is natural to relate two such min-subgroups. P. Das introduced in [5] the following relation between min-subgroups of a group.
Definition 4.
Let G be a group and μ , η two min-subgroups of G. We say that μ is equivalent to η, written μ η , if μ t t μ ( G ) = η s s η ( G ) where μ ( G ) and η ( G ) are the ranges of μ and η, respectively. The class of an element μ will be denoted by [ μ ] .
There are other significant equivalences on min-subgroups [26,27,28]. A study on their connections has been recently presented in [29]. Our paper focuses only on the given one by P. Das, which is the most relevant in the literature. Many results can be transferred easily taking into account the implications diagram from [29]. A. Jain characterized the equivalence relation ∼ as follows.
Proposition 4
([30]). Let G be a group and μ , η two min-subgroups of G. The following assertions are equivalent:
1 .
μ ( x ) > μ ( y ) if and only if η ( x ) > η ( y ) .
2 .
μ ( x ) μ ( y ) if and only if η ( x ) η ( y ) .
3 .
{ μ t } t μ ( G ) = { η s } s η ( G ) .
4 .
{ μ t } t μ ( G ) = { η s } s η ( G ) .
We introduce the following example showing equivalence classes according to ∼ in order to illustrate how self-aggregation acts on the equivalence class.
Example 2.
Consider the min-subgroups μ , η , ν , σ and the aggregation A presented in Example 1. We have:
[ σ ] [ μ ] = [ ν ] [ η ]   and   [ σ ] [ η ] .
Moreover, self-aggregating each of these min-subgroups through A provides:
[ A ( μ , μ ) ] = [ μ ]
[ A ( η , η ) ] [ η ]
[ A ( ν , ν ) ] [ ν ]
[ A ( σ , σ ) ] = [ σ ]
The example shows that self-aggregation does not preserve equivalence classes in general. We dedicate the last part of the section to finding conditions on an aggregation function A, which ensures that a min-subgroup and its self-aggregation by A belong to the same equivalence class.
The following result is a straightforward consequence of Proposition 3.
Proposition 5.
If A is an averaging aggregation function and μ a min-subgroup of a group G, then [ A ( μ , , μ ) ] = [ μ ] .
The next proposition shows the relevance of jointly strictly monotone aggregation functions.
Proposition 6.
Let G be a group and A : [ 0 , 1 ] n [ 0 , 1 ] be an aggregation function. If A is jointly strictly monotone, then [ A ( μ , , μ ) ] = [ μ ] for each min-subgroup μ of G.
Proof. 
We need to prove that A ( μ , , μ ) and μ induce the same level sets. We will use the characterization of the Proposition 4. Let us take x , y G . Firstly, assume that μ ( x ) μ ( y ) ; by monotonicity of A, we have that A ( μ , , μ ) ( x ) A ( μ , , μ ) ( y ) .
Conversely, assume that A ( μ , , μ ) ( x ) A ( μ , , μ ) ( y ) . We must check that μ ( x ) μ ( y ) . By contradiction, μ ( x ) < μ ( y ) . Since A is jointly strictly monotone, we conclude that A ( μ ( x ) , , μ ( x ) ) < A ( μ ( y ) , , μ ( y ) ) ; equivalently, A ( μ , , μ ) ( x ) < A ( μ , , μ ) ( y ) , obtaining the desired contradiction. □
We proceed with the main result of the article. Let us recall that an aggregation function A is strictly increasing on its diagonal if for each x , y [ 0 , 1 ] , satisfying x < y ; then
A ( x , , x ) < A ( y , , y ) .
Theorem 2.
Let G be a group and A : [ 0 , 1 ] n [ 0 , 1 ] be an aggregation function. The following assertions are equivalent:
(1)
A is a strictly increasing function on its diagonal.
(2)
A ( μ , , μ ) and μ induce the same level sets.
Proof. 
1 2 . We will use the characterization of the Proposition 4. Let us take x , y G . Assume that μ ( x ) μ ( y ) ; by monotonicity of A, we have that A ( μ , , μ ) ( x ) A ( μ , , μ ) ( y ) .
Conversely, assume that A ( μ , , μ ) ( x ) A ( μ , , μ ) ( y ) . We must check that μ ( x ) μ ( y ) . By contradiction, suppose that μ ( x ) < μ ( y ) . Since A is a strict increasing function on its diagonal, we conclude that A ( μ ( x ) , , μ ( x ) ) < A ( μ ( y ) , , μ ( y ) ) , and equivalently, A ( μ , , μ ) ( x ) < A ( μ , , μ ) ( y ) , which is a contradiction.
2 1 . We prove that if A is not strictly increasing on its diagonal, then there is a min-subgroup μ G such that A ( μ , , μ ) and μ do not have the same level sets. Under this premise, there are a , b [ 0 , 1 ] such that
a < b   and   A ( a , , a ) A ( b , , b ) .
By monotonicity, we have that A ( a , , a ) = A ( b , , b ) . Let us create the fuzzy set μ : G [ 0 , 1 ] , satisfying μ ( e ) = b and μ ( x ) = a whenever x e . (We remember that e denotes the neutral element of G.) Clearly, μ is a min-subgroup of G according to Proposition 1. Therefore, considering an element x e , we conclude that
A ( μ ( x ) , , μ ( x ) ) = A ( a , , a ) = A ( b , , b ) = A ( μ ( e ) , , μ ( e ) ) .
Since μ ( x ) < μ ( e ) , they induce different level sets. □
As a direct consequence of the previous theorem, we have obtained the desired characterization.
Corollary 1.
Let μ be a min-subgroup of a group G. If A is a strict t-norm or a strict t-conorm, then A ( μ , , μ ) belongs to the same equivalence class as μ.

5. Concluding Remarks

Let A be a generic aggregation function, G a group, μ a min-subgroup of G, and [ μ ] the Das class of μ .
Firstly we have shown that the structure of min-subgroup is preserved by arbitrary self-aggregation functions—i.e., A ( μ , , μ ) is a min-subgroup—and we have studied when A ( μ , , μ ) is comparable to μ .
Secondly, we have shown an example of an aggregation function A and a fuzzy subgroup μ satisfying [ A ( μ , , μ ) ] [ μ ] . Hence, the Das class of a min-subgroup is not necessarily preserved by an arbitrary aggregation function. We have shown that this class is preserved if A is an averaging or a jointly strictly monotonous aggregation function.
Thirdly, our main results states that A ( μ , , μ ) and μ induce the same level sets if and only if A is a strictly increasing function on its diagonal. This result implies that if A is a strict t-norm or a strict t-conorm, A ( μ , , μ ) belong to the same equivalence class as μ .
Future research could examine under what conditions the Lukasiewicz and product subgroup structures are preserved by arbitrary self-aggregation functions and explore the implications of the migrativity property ([31]) for the preservation of these subgroup structures under self-aggregation functions.

Author Contributions

Investigation, C.B., S.A.-T. and J.E.; writing—original draft preparation, C.B., S.A.-T. and J.E.; writing—review and editing, C.B., S.A.-T. and J.E. All authors have read and agreed to the published version of the manuscript.

Funding

The authors acknowledge the financial support of the Spanish Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and the European FEDER funds (Grant MTM 2016-79422-P and Grant PGC 2018-098623-B-I00). C. Bejines has also been supported by the Ministry of Education of the Czech Republic under Project GACR 19-09967S.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The authors would like to express their gratitude towards María Jesús Chasco for her support and collaboration during all these years.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

References

  1. Beliakov, G.; Pradera, A.; Calvo, T. Aggregation Functions: A Guide for Practitioners; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
  2. Beliakov, G.; Sola, H.B.; Sánchez, T.C. A Practical Guide to Averaging Functions; Springer: Heilderberg, Germany; New York, NY, USA; Dordrech, The Netherlands; London, UK, 2016. [Google Scholar]
  3. Calvo, T.; Kolesárová, A.; Komorníková, M.; Mesiar, R. Aggregation Operators; Physica-Verlag: Heidelberg, Germany, 2002; pp. 3–104. [Google Scholar]
  4. Rosenfeld, A. Fuzzy groups. J. Math. Anal. Appl. 1971, 35, 512–517. [Google Scholar] [CrossRef] [Green Version]
  5. Das, P. Fuzzy groups and level subgroups. J. Math. Anal. Appl. 1981, 84, 264–269. [Google Scholar] [CrossRef] [Green Version]
  6. Anthony, J.; Sherwood, H. Fuzzy groups redefined. J. Math. Anal. Appl. 1979, 69, 124–130. [Google Scholar] [CrossRef] [Green Version]
  7. Formato, F.; Gerla, G.; Scarpati, L. Fuzzy subgroups and similarities. Soft Comput. 1999, 3, 1–6. [Google Scholar] [CrossRef] [Green Version]
  8. Boixader, D.; Recasens, J. On the relationship between fuzzy subgroups and indistinguishability operators. Fuzzy Sets Syst. 2019, 373, 149–163. [Google Scholar] [CrossRef]
  9. Bloch, I.; Maître, H. Fuzzy mathematical morphologies: A comparative study. Pattern Recognit. 1995, 28, 1341–1387. [Google Scholar] [CrossRef]
  10. Elorza, J.; Fuentes-González, R.; Bragard, J.; Burillo, P. On the relation between fuzzy closing morphological operators, fuzzy consequence operators induced by fuzzy preorders and fuzzy closure and co-closure systems. Fuzzy Sets Syst. 2013, 218, 73–89. [Google Scholar] [CrossRef]
  11. Soille, P. Morphological Image Analysis: Principles and Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
  12. Bejines, C.; Chasco, M.; Elorza, J. Aggregation of fuzzy subgroups. Fuzzy Sets Syst. 2021, 418, 170–184. [Google Scholar] [CrossRef]
  13. Bejines, C.; Ardanza-Trevijano, S.; Chasco, M.; Elorza, J. Aggregation of indistinguishability operators. Fuzzy Sets Syst. 2021. [Google Scholar] [CrossRef]
  14. Drewniak, J.; Dudziak, U. Preservation of properties of fuzzy relations during aggregation processes. Kybernetika 2007, 43, 115–132. [Google Scholar]
  15. Dudziak, U. Preservation of t-norm and t-conorm based properties of fuzzy relations during aggregation process. In Proceedings of the EUSFLAT Conference, Milano, Italy, 11–13 September 2013. [Google Scholar]
  16. Fodor, J.C.; Roubens, M. Fuzzy Preference Modelling and Multicriteria Decision Support; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1994; Volume 14. [Google Scholar]
  17. He, X.; Li, Y.; Qin, K.; Meng, D. On the TL-transitivity of fuzzy similarity measures. Fuzzy Sets Syst. 2017, 322, 54–69. [Google Scholar] [CrossRef]
  18. Jacas, J.; Recasens, J. Aggregation of T-transitive relations. Int. J. Intell. Syst. 2003, 18, 1193–1214. [Google Scholar] [CrossRef]
  19. Mayor, G.; Recasens, J. Preserving T-Transitivity. In Artificial Intelligence Research and Development; Nebot, A., Binefa, X., de Mantaras, L., Eds.; IOS Press: Amsterdam, The Netherlands; Berlin, Germany; Washington, DC, USA, 2016; pp. 79–87. [Google Scholar]
  20. Pedraza, T.; Rodríguez-López, J.; Valero, Ó. Aggregation of fuzzy quasi-metrics. Inf. Sci. 2020. [Google Scholar] [CrossRef]
  21. Saminger, S.; Mesiar, R.; Bodenhofer, U. Domination of aggregation operators and preservation of transitivity. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 2002, 10, 11–35. [Google Scholar] [CrossRef]
  22. Klement, E.P.; Mesiar, R.; Pap, E. Triangular Norms; Springer Science & Business Media: Dordrecht, The Netherlands, 2013; Volume 8. [Google Scholar]
  23. Zadeh, L. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
  24. Dixit, V.; Kumar, R.; Ajmal, N. Level subgroups and union of fuzzy subgroups. Fuzzy Sets Syst. 1990, 37, 359–371. [Google Scholar] [CrossRef]
  25. Murali, V.; Makamba, B. On an equivalence of fuzzy subgroups I. Fuzzy Sets Syst. 2001, 123, 259–264. [Google Scholar] [CrossRef]
  26. Murali, V.; Makamba, B. On an equivalence of fuzzy subgroups III. Int. J. Math. Math. Sci. 2003, 2003, 2303–2313. [Google Scholar] [CrossRef] [Green Version]
  27. Ray, S. Isomorphic fuzzy groups. Fuzzy Sets Syst. 1992, 50, 201–207. [Google Scholar] [CrossRef]
  28. Zhang, Y. Some properties on fuzzy subgroups. Fuzzy Sets Syst. 2001, 119, 427–438. [Google Scholar] [CrossRef]
  29. Bejines, C.; Chasco, M.J.; Elorza, J.; Montes, S. Equivalence relations on fuzzy subgroups. In Proceedings of the Conference of the Spanish Association for Artificial Intelligence, Granada, Spain, 23–26 October 2018; pp. 143–153. [Google Scholar]
  30. Jain, A. Fuzzy subgroups and certain equivalence relations. Iran. J. Fuzzy Syst. 2006, 3, 75–91. [Google Scholar]
  31. Bustince, H.; De Baets, B.; Fernández, J.; Mesiar, R.; Montero, J. A generalization of the migrativity property of aggregation functions. Inf. Sci. 2012, 191, 76–85. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Bejines, C.; Ardanza-Trevijano, S.; Elorza, J. On Self-Aggregations of Min-Subgroups. Axioms 2021, 10, 201. https://doi.org/10.3390/axioms10030201

AMA Style

Bejines C, Ardanza-Trevijano S, Elorza J. On Self-Aggregations of Min-Subgroups. Axioms. 2021; 10(3):201. https://doi.org/10.3390/axioms10030201

Chicago/Turabian Style

Bejines, Carlos, Sergio Ardanza-Trevijano, and Jorge Elorza. 2021. "On Self-Aggregations of Min-Subgroups" Axioms 10, no. 3: 201. https://doi.org/10.3390/axioms10030201

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop