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Fuzzy groups, fuzzy functions and fuzzy equivalence relations

# Fuzzy groups, fuzzy functions and fuzzy equivalence relations,10.1016/S0165-0114(03)00301-4,Fuzzy Sets and Systems,Mustafa Demirci,Jordi Recasens

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 μ in such a way that it can be interpreted as the fuzzy quotient group X/μ.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.
Journal: Fuzzy Sets and Systems - FSS , vol. 144, no. 3, pp. 441-458, 2004
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## Citation Context (6)

• ...Indeed, M. Demirci and J. Recasens proposed in [9] a fuzzy subgroup to be a group (X; ~ ), where ~ (a;b;c) denotes the degree to which c = a b. An even more fuzzier algebraic structure is one were both the elements belongs to the underlying set to a degree and the result of the operation is the \real" result up to some degree...

### Apostolos Syropoulos. Fuzzy Chemical Abstract Machines

• ...1041-4347/08/\$25.00 2008 IEEE Published by the IEEE Computer Society see [27], [28], [29], [30], and [31]...

### Robert Nowicki. On Combining Neuro-Fuzzy Architectures with the Rough Set Theory to So...

• ...A conceptual origin of our proposed MFF-LSE may be found in Sasaki [14], Demirci [8]-[9], Demirci and Recasens [10] discusses the general properties of “Fuzzy Functions” from a perspective of mathematical theory...

### B. Turksen. Mixed Fuzzy Functions

• ...Although there are many variations of fuzzy functions in the literature [16],[17],[46], a conceptual origin of our proposed FF-LSE may be found in Demirci [12]-[14] where the general properties of “Fuzzy Functions” from the perspective of mathematical theory is discussed...

### I. Burhan, et al. Comparison of Fuzzy Functions with Fuzzy Rule Base Approaches

• ...For other works on combining the rough set and fuzzy set theories see [2], [3], [4], [6] and [11]...

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