Fuzzy groups, fuzzy functions and fuzzy equivalence relations

To each fuzzy subgroup a fuzzy equivalence relation is associated and it is proved that a fuzzy subgroup is normal if and only if the operation of the group is compatible with its associated fuzzy equivalence relation. In the definition of fuzzy subgroup only the subset is fuzzy whilst the group operation remains crisp. In an opposite direction, another structure can be constructed maintaining crisp the basic set X of the group and fuzzifying the group operation. When in addition it is imposed to the fuzzy operation to be compatible with a given fuzzy equivalence relation, this structure is called a vague group. The results of the paper allow us to associate a vague group to every fuzzy subgroup (mu) in such a way that it can be interpreted as the fuzzy quotient group X/(mu). Special attention is paid to normal fuzzy groups and vague groups of the additive group (R,+) of the real numbers. Two examples related to triangular numbers illustrating the results of the paper are provided.