Advertisement

EUSFLAT 2017, IWIFSGN 2017: Advances in Fuzzy Logic and Technology 2017 pp 159-167

# On the Preservation of an Equivalence Relation Between Fuzzy Subgroups

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)

## Abstract

Two fuzzy subgroups $$\mu ,\eta$$ of a group G are said to be equivalent if they have the same family of level set subgroups. Although it is well known that given two fuzzy subgroups $$\mu ,\eta$$ of a group G their maximum is not always a fuzzy subgroup, it is clear that the maximum of two equivalent fuzzy subgroups is a fuzzy subgroup. We prove that the composition of two equivalent fuzzy subgroups by means of an aggregation function is again a fuzzy subgroup. Moreover, we prove that if two equivalent subgroups have the sup property their corresponding compositions by any aggregation function also have the sup property. Finally, we characterize the aggregation functions such that when applied to two equivalent fuzzy subgroups, the obtained fuzzy subgroup is equivalent to both of them. These results extend the particular results given by Jain for the maximum and the minimum of two fuzzy subgroups.

## Keywords

t-norm t-conorm Aggregation function Fuzzy subgroup Level fuzzy subset Strong level fuzzy subset Sup property Equivalent fuzzy subgroups

## References

1. 1.
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelberg (2007)
2. 2.
Calvo, T., Mayor, G., Mesiar, R.: Aggregation Operation. Aggregation operation. Physica-Verlag, Heidelberg (2002)
3. 3.
Das, P.S.: Fuzzy groups and level subgroups. J. Math. Anal. Appl. 84, 264–269 (1981)
4. 4.
Fodor, J., Kacprzyk, J.: Aspect of Soft Computing, Intelligent robotics and Control. Studies in Computational Intelligence, vol. 241. Springer, Heidelberg (2009)
5. 5.
Gupta, M.M., Qi, J.: Theory of T-norms and fuzzy inference methods. Fuzzy Sets Syst. 40, 431–450 (1991)
6. 6.
Jain, A.: Fuzzy subgroup and certain equivalence relations. Iran. J. Fuzzy Syst. 3, 75–91 (2006)
7. 7.
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)
8. 8.
Menger, K.: Statistical metrics. In: Proceedings of N.A.S.H, vol. 28, pp. 535-537 (1942)Google Scholar
9. 9.
Murali, V., Makamba, B.: On an equivalence of fuzzy subgroups I. Fuzzy Sets Syst. 123, 259–264 (2001)
10. 10.
Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl. 35, 512–517 (1971)
11. 11.
Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Math. Debrecen 8, 169–186 (1961)

## Copyright information

© Springer International Publishing AG 2018

## Authors and Affiliations

• Carlos Bejines
• 1
Email author
• María Jesús Chasco
• 1
• Jorge Elorza
• 1
• Susana Montes
• 2
1. 1.Dto de Física y Matemática AplicadaUniversidad de NavarraPamplonaSpain
2. 2.Dto de Estadística e Investigación OperativaUniversidad de OviedoOviedoSpain