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 de-nition 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 in such a way that it can be interpreted as the fuzzy quotient group X= . 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. c © 2003 Elsevier B.V. All rights reserved.