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On the Preservation of an Equivalence Relation Between Fuzzy Subgroups

  • Carlos BejinesEmail author
  • María Jesús Chasco
  • Jorge Elorza
  • Susana Montes
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 

Notes

Acknowledgement

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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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

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