eu FUZZY NORMAL CONGRUENCES AND FUZZY COSET RELATIONS ON GROUPS

L.A. Zadeh [11] introduced the concept of fuzzy set,.and since then there has been a tremendous interest in the subject due to its diverse applications ranging from engineering and computer science to social behavior studies. This concept is very general and many researchers applied this subject to the various structures and obtained some useful results in their areas. A. Rosenfeld [7] formulated the concept of a fuzzy subgroup and showed how some basic notions of group theory should be extended in an elementary manner to develop the theory of fuzzy groups. J.P. Kim and D.R. Bae [3] characterized the composition of two congruence relations in semigroups and groups. They also defined a fuzzy congruence class and studied this property. N.P. Mukherjee


Introduction
L.A. Zadeh [11] introduced the concept of fuzzy set,.andsince then there has been a tremendous interest in the subject due to its diverse applications ranging from engineering and computer science to social behavior studies.
This concept is very general and many researchers applied this subject to the various structures and obtained some useful results in their areas.A. Rosenfeld [7] formulated the concept of a fuzzy subgroup and showed how some basic notions of group theory should be extended in an elementary manner to develop the theory of fuzzy groups.J.P. Kim and D.R. Bae [3] characterized the composition of two congruence relations in semigroups and groups.They also defined a fuzzy congruence class and studied this property.N.P. Mukherjee [5] introduced and studied the concepts of fuzzy normal subgroups and fuzzy cosets in groups.He proved that µ is a fuzzy normal subgroup of finite group G if and if the level subgroups of µ are normal subgroups of G.
In this work, by deep view, we extend the concepts of fuzzy normal subgroups and fuzzy cosets to the fuzzy relation and introduce the concepts of fuzzy normal congruences and fuzzy coset relations in groups.We also define the operation ⊙ on fuzzy relations and show that the family of all fuzzy congruences with this operation in commutative semigroup is a semigroup and clarify that the set of all fuzzy coset relations obtained from fuzzy normal congruences with operation ⊙ is a group.

Preliminaries
Note.Hereinafter, G under operation * is always a group unless otherwise specified.
(ii) Fuzzy equivalence relation α on G is called a fuzzy congruence if α(x * t, y * z) ≥ α(x, y) ∧ α(t, z).Proposition 2.5.[3] Let P and Q be fuzzy congruences on semigroup S. Then: Proposition 2.6.[3] Let P and Q be fuzzy congruences on group G. Then: (i) is the family of all fuzzy congruences of a group G.
Lemma 2.7.[4] Let α be a fuzzy congruence on group G. Then: Here x, y, z, a, b ∈ G.
Proposition 2.8.[10] Let µ be a fuzzy ideal of semigroup S. Then the fuzzy relation θ µ as follows: is a fuzzy congruence on S.

On Fuzzy Congruence on Groups
Example 3.1.Let G = Z 4 .Then fuzzy relation  ), is a fuzzy congruence on G.
Proof.It is clear that α is a fuzzy reflexive and symmetric.To transitivity it is sufficient to prove, if α(x, t) = α(t, y) = 1, then α(x, y) = 1.Therefore This completes the proof.Definition 3.4.Let (G, * ) be a semigroup and α and β be fuzzy relations on G. Then the composition α ⊙ β is defined as follows: Proof.Suppose that α and β are fuzzy congruences on commutative semigroup G. Then by Theorem 3.5, it is sufficient to prove that α ⊙ β is a fuzzy congruence on G.
By associative and commutative property of operation ⊙ we have Definition 3.7.Let A, B, C and D be subsets of semigroup S, α be a fuzzy relation on S and f and g be morphisms from A to B and C to D respectively .Then: (ii) f and g are called f-g one to one if f (x) = g(y) =⇒ x = y.Theorem 3.8.Let f and g be homomorphisms from G onto H, G and H be semigroups (groups), f and g be f-g one to one and α be a f × g-invariant fuzzy congruence on G. Then By the definition it is clear that f × g(α) is symmetric.If x, y, z, t ∈ H, then Corollary 3.9.(i)If f and g are isomorphisms from G onto H, f and g are f-g one to one and α is a fuzzy congruence on G, then (f × g)(α) is a fuzzy congruence on H. (ii) If f is a isomorphism from G onto H and α is a fuzzy congruence on G, then (f × f )(α) is a fuzzy congruence on H. Theorem 3.10.Let f be a homomorphism from G into H and θ be a fuzzy congruence on H. Then is a fuzzy congruence on G.
Proof.The proof is straightforward.

Theorem 3.11. Let f be an isomorphism from G onto H and α and θ be fuzzy congruences on G and H respectively. Then
Proof.By Corollary 3.9 and Theorem 3.10, the proof is easy.

Fuzzy Normal Congruences
Definition 4.1.Let α be a fuzzy congruence on semigroup G. Then α is called a fuzzy normal congruence on G, if α(x * t, y * z) = α(t * x, z * y) f or all x, y, t, z ∈ G Lemma 4.2.Let α be a fuzzy congruence on G and x, y, z, t ∈ G. Then: The proof is obvious.Lemma 5.2.Let α be a fuzzy normal congruence, α (a,b) and α (c,d) be fuzzy coset relations on G. Then By the similar way we have α b) and α (c,d) are fuzzy coset relations, then the composition α (a,b) ⊙ α (c,d) is defined as follows: ( Proof.First, we prove the operation ⊙ is well-defined.Suppose that α (a,b) = α (c,d) and α (e,f ) = α (g,h) .Then Therefore α (a,b) ⊙ α (e,f ) = α (c,d) ⊙ α (g,h) .For all x 1 , x 2 , y 1 , y 2 ∈ V such that x = x 1 * x 2 and y = y 1 * y 2 we have Proof.Since G is a group and by Proposition 5.4, (G × G) α is closed under the operation ⊙.

Conclusion
We know the equivalence classes induced by a congruence relation to study an algebraic system has a vital role.For instance the theory of rough sets and approximation sets has close relationship with congruence relation and equivalence classes.Similarly fuzzy congruence relations have this role in study of fuzzy algebraic systems.In this work we defined the concepts of fuzzy normal congruences and fuzzy coset relations in semigroups and groups.In a natural way the level subset of a fuzzy normal congruence must be a normal congruence that so far has not defined and we tend to study this concept in other work.

5. Fuzzy Coset Relations Definition 5 . 1 .
Let α be a fuzzy congruence on G. Then for a, b ∈ G, the fuzzy relation α (a,b) on G defined by α (a,b) (x, y) = α(a * x −1 , b * y −1 ), for all x, y ∈ G, is called a fuzzy coset relation determined by a, b and α.