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.
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References
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelberg (2007)
Calvo, T., Mayor, G., Mesiar, R.: Aggregation Operation. Aggregation operation. Physica-Verlag, Heidelberg (2002)
Das, P.S.: Fuzzy groups and level subgroups. J. Math. Anal. Appl. 84, 264–269 (1981)
Fodor, J., Kacprzyk, J.: Aspect of Soft Computing, Intelligent robotics and Control. Studies in Computational Intelligence, vol. 241. Springer, Heidelberg (2009)
Gupta, M.M., Qi, J.: Theory of T-norms and fuzzy inference methods. Fuzzy Sets Syst. 40, 431–450 (1991)
Jain, A.: Fuzzy subgroup and certain equivalence relations. Iran. J. Fuzzy Syst. 3, 75–91 (2006)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)
Menger, K.: Statistical metrics. In: Proceedings of N.A.S.H, vol. 28, pp. 535-537 (1942)
Murali, V., Makamba, B.: On an equivalence of fuzzy subgroups I. Fuzzy Sets Syst. 123, 259–264 (2001)
Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl. 35, 512–517 (1971)
Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Math. Debrecen 8, 169–186 (1961)
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The authors acknowledge to the referee for your successful suggestions and advices.
The authors acknowledge the financial support of the Spanish Ministerio de Economía y Competitividad (Grant TIN2014-59543-P and Grant MTM 2016-79422-P) and Carlos Bejines also thanks the support of the Asociación de Amigos of the University of Navarra.
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Bejines, C., Chasco, M.J., Elorza, J., Montes, S. (2018). On the Preservation of an Equivalence Relation Between Fuzzy Subgroups. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 641. Springer, Cham. https://doi.org/10.1007/978-3-319-66830-7_15
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DOI: https://doi.org/10.1007/978-3-319-66830-7_15
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