TRANSLATES OF VAGUE NORMAL GROUPS

In this paper we introduce translate operators on vague cut set, vague normal groups, and studied their properties. These concepts are used in the development of some important results and theorems about translates of vague homomorphism, vague normal groups. Also some of their important properties have been investigated. AMS Subject Classification: 08A72, 20N25, 03E72

by means of a membership function µ A defined from X into [0, 1] has revolutionized the theory of Mathematical modeling decision making etc.In handling the imprecise real life situations mathematically.Now several branches of fuzzy mathematics like fuzzy algebra, fuzzy topology, fuzzy control theory, fuzzy measure theory etc. have emerged.But in the decision making, the fuzzy theory takes care of membership of an element x only, that is the evidence against x belonging to A. It is felt by several decision makers and researchers that in proper decision making, the evidence belongs to A and evidence not belongs to A are both necessary and how much X belongs to A or how much x does not belongs to A are necessary.
Several generalizations of Zadeh.L. A'sfuzzy set theory have been proposed, such as L-fuzzy sets (see [4]), Interval valued fuzzy sets.Intuitionistic fuzzy sets by K.T. Atanassov [1] and W.L. Gahu, D.J. Buehrer Vague sets, see [3], are mathematically equivalent.Any such set A of a given Universe X can be characterized by means of a pair of function (t A , f A ) where t A and f A are function from X in to [01] such that 0 ≤ t A (x) + f A (x) ≤ 1 for all x in X.The set t A is called the truth function and the set f A is called false function or non membership function and t A (x) gives the evidence of how much x ∈A f A (x) gives the evidence of how much x does not ∈ A. These concepts are being applied in several areas like decisionmaking, fuzy control, knowledge discovery and fault diagonsis etc.I t is belived the vague sets (or equivalently instuitionisticfuzzy sets) will more useful in decision making, and other areas of Mathematical modeling.Through K.T. Atanassov [1] instuitionistic fuzzysets, W.L. Gau and D.J. Buehrer [3] and some other areas of Mathematical modeling.Since then the theory fuzzy sets developed extensivelyand embraced almost all subjects like engineering science and Technology.But the membership function µ A gives only a approximation belong to A. To avid this and obtain a better estimation and analysis of data decision making.W.L. Gau and D.J. Bueher in [3] have initiated the study of vague sets with the hope that they form a better tool to understand, interpret and solve real life problems which are in general vague, than the theory of Fuzzy sets do.Ranjit Biswas [9] initiated the study of vague groups and N. Ramakrishna [8], [10], [11], T. Eswarlal [2], [8], [10] create extended the study of vague algebra.The objective of this paper is to contribute further to the study of vague algebra by introducing concepts of translate operators on vague cut set, translate operators on vague homomorphism and translate operators on vague normal groups.
Here the element xy stands for x * y.
Definition 1.4.Let A be a vague set of a universe G with true-membership function t A , and false membership function f A .For α, β ǫ[0, 1] with α ≤ β, the (α, β) cut or vague cut of a vague set A is the crisp subset of G is given by Definition 1.5.The α-cut, A α of the vague set A is the (α, α)cut of A, and hence given by A α = {x|xǫG, t A (x) ≥ α }.Definition 1.6.Let A be a Vague group of a group G Then A is Called Vague normal group if for all x, y ∈ G, V A (xy) = V A (yx).Alternatively, we can say that, a vague groupA is said to be vague normal group of G if V A (x) = V A (yxy −1 ) for all x, y ∈ G.
We now introduce the following definitins.Definition 1.7.Let A be a Vague group of a group G then the set N(A) = {a ∈ G|V A (axa −1 ) = V A (x) for all a, ∈ G} is called vague normalizer of A. Definition 1.8.Let A be a vague group of a group G Then the set

for all i, and hence
A (α,β) (P i ).
Theorem 2.6.If P is vague group of G. Then T θ+ (A (α,β) (P )) and T θ− (A (α,β) (P )) are a translate vague group of G, where t P (e) ≥ α, f P (e) ≤ β and e is the identity element of G.
Proof.Let P is vague group of G.A (α,β) (P ) is vague cut-set.We shall show that T θ+ (A (α,β) (P )) is a translate of vague group of G.
Similarly, we can show that T θ− (A (α,β) (P )) is a vague group of G.
Theorem 2.7.P and Q are two translates of vague groups of G thenP ∩ Q is translates of vague group of G.
Proof.P and Q are two translates of vague groups of G. Then T θ+(P ∩Q) is a translates of vague group of G. Since the result hods true.Now Hence From ( 1) and ( 2): T θ+(P ∩Q) is a translates of vague group of G.
Theorem 2.8.Let A be a translates of vague group of G. Then α-cut A α is a translates of vague group of G.
From ( 1) and (2) A α cut set is a translates of vague group of G.

Translate Operators on Vague Homomorphism Groups and
Vague Normal Groups Defination 3.1.Let G be a vague normal group of be a vague set P then the translate operators of Vague normal group can be defined by and Theorem 3.2.If P is a translate of vague normal group of G. Then T θ+ (A (α,β) (P )) is a translate of vague normal group of G. where t P (e) ≥ α, f P (e) ≤ β and e is the identity element of G.
Proof.Given that P is a vague normal group of G ie V P (xy) = V P (yx) f orallx ∈ Giet P (xy) = t P (yx)and f P (xy) = f P (yx) for all x, y ∈ G. Now we shall that T θ+ (A (α,β) (P )) is Translate of vague normal group of G, i.e.
Theorem 3.3.Let G and G 1 be any two vague groups then the homomorphic image of translates of vague group A of G is a vague group of G 1 .
Proof.Let G and G 1 be any two groups and Φ : G → G 1 be a homomorphism.
Therefore Φ(x, y) = Φ(x).Φ(y) for all x, y ∈ G. Let V = Φ(T α+ ) be a translates of a vague group of G.We shall show that V = Φ(T α+ ) is a vague group of G 1 .
Theorem 3.4.Let G and G 1 be any two vague groups.Then the homomorphic image of an translates of vague normal group A of G is an vague normal group of G 1 .
Proof.Let G and G 1 be any two groups and Φ : G → G 1 be a homomorphism.
Let V = Φ(T α+ ) where T α+ is a translation of vague normal group of A of G.
We shall show that V = Φ(T α+ ) is a translates of vague normal group G 1 .For any Φ(x), Φ(y)inG Therefore (P Q) is translates of vague normal group of G. Theorem 3.9.Let P be a translates of vague group of G. Then K = {x ∈ G : t P (x) = t P (e)andf P (x) = f P (e)} is a translates of vague normal group of N (P ).
Proof.Let x ∈ K and y ∈ K ⇒ t P (x) = t P (e)andf P (x) = f P (e)}.consider Therefore f (P Q)T α+ (xyx −1 ) = N (P ), K = {x ∈ G : t P (x) = t P (e), and f P (x) = f P (e)} is a translates of vague normal group of N (P ).
Theorem 3.10.Let P be a translates vague group of a group G, and K={x ∈ G|V P (x) = V P (e)} is translates of vague normal group of G. Then K ⊆ C(P ).
Proof.Let x ∈ K. Then V P (x) = V P (e) ⇒ t P (x) = t P (e) and f P (x) = f P (e)for all x, y ∈ G.