On the Preservation of an Equivalence Relation Between Fuzzy Subgroups

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.