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INTUITIONISTIC Q-FUZZY HX GROUP
V. Lakshmana Gomathi Nayagam
Department of Mathematics, National Instituite of Technology, Thiruchirappalli-620 015, Tamilnadu, India
E-mail : [email protected]; [email protected]
R. Muthuraj*
Department of Mathematics, H. H. The Rajah’s College, Pudukkottai – 622 001, Tamilnadu, India
E-mail : [email protected]
K. H. Manikandan
Department of Mathematics, PSNA College of Engineering and Technology, Dindigul-624 622, Tamilnadu, India
E-mail : [email protected]
(Received on: 29-06-11; Accepted on: 10-07-11)
--- ABSTRACT
I
n this paper, we define the concept of an intuitionistic Q-fuzzy HX group and define a new algebraic structure of intuitionistic Q-fuzzy HX group and some related properties are investigated. We also define the level subsets of an intuitionistic Q-fuzzy HX group and discussed some of its properties.Keywords: Intuitionistic fuzzy set, intuitionistic Q-fuzzy set, intuitionistic fuzzy subgroup, intuitionistic Q-fuzzy subgroup, intuitionistic Q-fuzzy HX subgroup and level sub HX groups.
AMS Subject Classification (2000): 20N25, 03E72, 03F055, 06F35, 03G25.
---1. INTRODUCTION
K.H.Kim [2] introduced the concept of intuitionistic Q-fuzzy semi prime ideals in semi groups and Osman kazanci, sultan yamark and serife yilmaz [10] introduced the concept of intuitionistic Q-fuzzy R-subgroups of near rings. A.Solairaju and R.Nagarajan [13][14] introduced and defined a new algebraic structure of Q-fuzzy groups. Li Hongxing [5] introduced the concept of HX group and the authors Luo Chengzhong , Mi Honghai , Li Hongxing [6] introduced the concept of fuzzy HX group. In this paper we define a new concept of intuitionistic Q-fuzzy HX group and its level sub HX groups and study some of their related properties.
2. PRELIMINARIES
In this section, we site the fundamental definitions that will be used in the sequel. Throughout this paper, G = (G , *) is a group, e is the identity element of G, and xy, we mean x * y.
2.1 Definition [1]: An intuitionistic fuzzy subset ( IFS ) in a set X is defined as an object of the form ={ x , µ(x) , ν(x) / x∈X }, where µ : X → [0,1] and ν : X → [0,1] define the degree of membership and the degree of non-membership of the element x∈X respectively to be in and for every x∈X, 0 ≤ µ(x) + ν(x) ≤ 1.
2.2 Definition [2]: An intuitionistic Q-fuzzy subset in a set X is defined as an object of the form ={ (x,q), µ(x,q), ν(x,q) / x∈X and q∈Q }, where µ : X × Q → [0,1] and ν : X × Q → [0,1] define the degree of membership and
R. Muthuraj* et al. / INTUITIONISTIC Q-FUZZY HX GROUP/ IJMA- 2(7), July-2011, Page: 1133-1139
the degree of non-membership of the element (x,q) ∈ X × Q respectively and for every (x,q) ∈ X × Q, 0 ≤ µ(x,q) + ν(x,q) ≤ 1.
2.3 Definition [5]: In 2G-{φ}, a nonempty set ϑ
⊂
2G-{φ} is called a HX group on G, if ϑ is a group with respect to the algebraic operation defined by AB = {ab / a ∈ A and b ∈ B}, and its unit element is denoted by E.3. INTUITIONISTIC FUZZY HX GROUPS
3.1 Definition: Let G be a group and a non-empty set ϑ
⊂
2G-{φ} is a HX group on G. An intuitionistic fuzzy subset = A, µ(A) , ν(A) of a HX group ϑ is said to be an intuitionistic fuzzy HX subgroup of a HX group ϑ if the following conditions are satisfied. For all A and B ∈ϑ,(i) µ(AB) ≥ min{ µ(A), µ(B) }, (ii) ν(AB) ≤ max{ ν(A), ν(B)) }. (iii) µ(A-1) = µ(A) ,
(iv) ν(A-1) = ν(A) .
3.1 Example: Let G = { 1 , -1 } ,then (G , • ) is a group and ϑ = { {1} ,{-1} }, then (ϑ , • ) is a Hx group. Define
µ : ϑ [0,1] by µ ( {1} ) = 0.8 and µ ( {-1} ) = 0.5. ν: ϑ [0,1] by ν ( {1} ) = 0.1 and ν ( {-1} ) = 0.3.
Clearly is an intuitionistic fuzzy HX group of a HX group ϑ .
3.2 Definition: An intuitionistic Q-fuzzy subset ={ (A,q) , µ(A,q) , ν(A,q) / A∈ϑ and q∈Q } of a HX group ϑ is said to be an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ if the following conditions are satisfied. For all A and B ∈ϑ and q∈Q,
(i) µ(AB,q) ≥ min{ µ(A,q), µ(B,q) }, (ii) ν(AB,q) ≤ max{ ν(A,q), ν(B,q)) }, (iii) µ(A-1,q) = µ(A,q) ,
(iv) ν(A-1,q) = ν(A,q) .
4. PROPERTIES OF INTUITIONISTIC Q-FUZZY HX SUBGROUP
In this section, we discuss some of the properties of intuitionistic Q-fuzzy HX subgroup.
4.1 Theorem: Let be an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ then
(i) µ (A, q) µ (E, q) and ν (A, q) ν (E, q) for all A ∈ϑ and q ∈Q and E is the identity element of ϑ.
(ii) The subset H = {A∈ϑ / µ (A, q) = µ (E, q) and ν (A, q) = ν (E, q)} is a sub HX group of ϑ.
Proof:
(i) Let A∈ϑ and q ∈Q,
µ (A, q) = min { µ (A, q ) , µ (A, q ) } = min { µ(A, q ) , µ(A-1 , q ) } µ (AA-1 , q )
= µ (E, q).
µ (A, q ) µ (E , q ) for all A ∈ϑ and q ∈Q.
ν (A, q) = max { ν (A, q ) , ν (A, q ) } = max { ν (A, q ) , ν (A-1 , q ) } ν (AA-1 , q )
= ν (E, q).
ν (A, q ) ν (E , q ) for all A ∈ϑ and q ∈Q.
R. Muthuraj* et al. / INTUITIONISTIC Q-FUZZY HX GROUP/ IJMA- 2(7), July-2011, Page: 1133-1139 Clearly H is non-empty as E ∈ H.
Let A, B ∈H and q ∈Q,
Then, µ (A, q) = µ (B, q) = µ (E, q) and ν (A, q) = ν (B, q) = ν (E, q).
µ (AB-1 , q) min {µ (A, q), µ (B-1 , q)} = min {µ (A, q), µ (B, q)}
= min {µ(E, q), µ (E, q)} = µ (E, q).
That is, µ (AB-1 ,q) µ (E, q) and by part i , µ (AB-1 , q) µ (E, q).
Hence, µ (AB-1, q) = µ (E, q) .
Also, ν (AB-1, q) max {ν (A, q), ν (B-1 , q)} = max {ν (A, q), ν (B, q)} = max {ν (E, q), ν (E, q)} = ν (E, q).
That is, ν (AB-1, q) ν (E, q) and by part i, ν (AB-1 , q) ν (E, q).
Hence, ν (AB-1, q) = ν (E, q).
Hence, µ (AB-1, q) = µ (E, q) and ν (AB-1, q) = ν (E, q) then AB-1 ∈H. Clearly, H is a sub HX group of ϑ.
4.2 Theorem: Let be any intuitionistic Q-fuzzy HX subgroup of a HX group ϑ with identity E. Then
µ (AB 1,q ) = µ (E, q ) µ (A, q ) = µ (B, q ) and
ν (AB 1,q ) = ν (E, q ) ν (A, q ) = ν (B, q ) , for all A,B in ϑ and q ∈Q .
Proof: Given is an intuitionistic Q-fuzzy HX subgroup of ϑ and µ (AB 1, q) = µ (E, q) and ν (AB 1,q ) = ν (E, q ) .
Then for all A , B in ϑ and q ∈Q ,
µ (A, q )= µ (A(B 1B), q ) = µ ((AB−1)B, q )
min { µ (AB−1, q ), µ (B, q )} = min { µ (E, q ) , µ (B, q )} = µ (B, q ).
That is, µ (A, q) µ (B, q).
That is, µ (AB 1, q) = µ (E, q) µ (A, q) µ (B, q).
Since, µ (BA 1, q) = µ ((BA 1)-1, q) = µ (AB 1, q) = µ (E, q).
That is, µ (BA 1,q ) = µ ( E, q ) , then µ (B, q ) µ (A, q ).
Hence, µ (A, q) = µ (B, q ).
ν (A, q ) = ν (A(B 1B), q ) = ν ((AB−1)B, q )
max { ν (AB−1, q ), ν (B, q )} = max { ν (E, q ) , ν (B, q )} = ν (B, q ).
R. Muthuraj* et al. / INTUITIONISTIC Q-FUZZY HX GROUP/ IJMA- 2(7), July-2011, Page: 1133-1139
That is, ν (AB 1, q) = ν (E, q) ν (A, q) ν (B, q).
Since, ν (BA 1, q) = ν ((BA 1)-1, q) = ν (AB 1, q) = ν (E, q).
That is, ν (BA 1,q ) = ν ( E, q ), then ν (B, q ) ν (A, q ).
Hence, ν (A, q) = ν (B, q).
4.3 Theorem: If is an intuitionistic Q-fuzzy HX subgroup of a HX group then for all A, B in ϑ and q ∈Q,
(i) µ (AB−1, q ) min {µ (A, q ) , µ (B, q )},
(ii) ν (AB−1, q ) max {ν (A, q ) , ν (B, q )}.
Proof: Let be an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ. Then for all A , B in ϑ and q ∈Q,
Now , µ (AB−1, q ) min{ µ (A, q ) , µ ( B−1, q )}. = min {µ (A, q), µ (B, q)}
µ (AB−1, q ) min{ µ (A, q ) , µ (B, q )}.
Now, ν (AB−1, q) max {ν (A, q ) , ν ( B−1, q )}. = max {ν (A, q ) , ν (B, q )}
ν (AB−1, q) max { ν (A, q ) , ν (B, q )}.
4.4 Theorem: If is an intuitionistic Q-fuzzy subset of a HX group and for all A , B in ϑ , q ∈Q,
(i) µ (AB−1, q ) min {µ (A, q ) , µ (B, q )},
(ii) ν (AB−1, q ) max {ν (A, q ) , ν (B, q )}, then is an intuitionistic Q-fuzzy HX subgroup of a HX group.
Proof: is an intuitionistic Q-fuzzy subset of a HX group and for all A, B in ϑ , q ∈Q,
(i) µ (AB−1, q ) min {µ (A, q ) , µ (B, q )},
Now, µ (A , q ) = µ (E(A−1) −1 , q ) min {µ (E, q ) , µ (A−1, q )} = µ (A−1, q )
Also, µ (A−1 , q ) = µ ( EA−1 , q ) min {µ (E, q ) , µ (A , q )} = µ (A , q ).
Hence, µ (A, q) = µ (A−1, q).
Also, µ (AB, q ) = µ (A(B−1) −1, q )
min {µ (A, q ) , µ (B−1, q )} min {µ (A, q ) , µ (B , q )}.
That is, µ (AB, q ) min {µ (A, q ) , µ (B , q )}.
(ii) ν (AB−1, q ) max {ν (A, q ) , ν (B, q )}.
Now, ν (A , q ) = ν (E(A−1) −1 , q ) max {ν (E, q ) , ν (A−1, q )} = ν (A−1, q ).
Also, ν (A−1 , q ) = ν ( EA−1 , q ) max {ν (E, q ) , ν (A , q )} = ν (A , q ).
Hence, ν (A, q) = ν (A−1, q).
Also, ν (AB, q) = ν (A (B−1) −1, q)
R. Muthuraj* et al. / INTUITIONISTIC Q-FUZZY HX GROUP/ IJMA- 2(7), July-2011, Page: 1133-1139 That is, ν (AB, q ) max {ν (A, q ) , ν (B , q )}.
Hence, is an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ.
5. PROPERTIES OF LEVEL SUBSETS OF AN INTUITIONISTIC Q-FUZZY HX SUBGROUP
In this section, we introduce the concept of level subset of an intuitionistic Q-fuzzy HX subgroup and discuss some of its properties.
5.1 Definition: Let be an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ. For any , ∈[0,1] , we define the set, < , > = { A ∈ϑ / µ(A , q ) and ν (A , q ) , for some q ∈Q }, is called the level subset of .
5.1 Theorem: Let be an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ and < , > is non- empty. Then , < , > is a sub HX group of ϑ.
Proof: For all A, B ∈ < , > and q ∈Q, we have,
µ (A, q ) and ν(A, q ) and µ (B, q ) and ν(B, q ) .
Now, µ (AB−1, q ) min {µ (A, q ), µ (B, q ) }.
µ (AB−1, q ) min { , } = . ν (AB−1, q ) max {ν (A, q ), ν (B, q ) }.
ν (AB−1, q ) max { , } = . AB−1 ∈ < , > .
Hence < , > is a sub HX group of ϑ.
5.2 Theorem: Let ϑ be a HX group and be a intuitionistic Q-fuzzy subset of ϑ such that < , > is a sub HX group of ϑ for all , ∈[0,1] then is an intuitionistic Q-fuzzy HX subgroup of ϑ.
Proof: Clearly is non-zero.
Let A, B ∈ϑ and q ∈Q .
Let µ (A, q) = 1 and ν(A, q ) = 1 and
µ (B, q ) = 2 and ν(B, q ) = 2.
Suppose let , 1 < 2 and 1 > 2 , then A, B ∈ < 1, 1 > .
As < 1, 1 > is a sub HX group of ϑ, AB−1∈ < 1, 1 > .
Hence, µ (AB−1, q) 1 = min { 1 , 2} min {µ (A, q ) , µ (B, q ) }
That is, µ (AB−1, q) min{µ (A, q), µ (B, q)}.
ν (AB−1, q ) 1 = max{ 1 , 2}
max {ν (A, q), ν (B, q)}
That is, ν (AB−1, q) max {ν (A, q), ν (B, q)}.
By Theorem 4.4, is an intuitionistic Q-fuzzy HX subgroup of ϑ.
5.2 Definition: Let be an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ. The sub HX groups < , > , , ∈[0,1] are called level sub HX groups of .
5.3 Theorem: An intuitionistic Q-fuzzy subset of ϑ is an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ if and only if the level subsets < , >, , ∈ Image , are sub HX groups of ϑ.
R. Muthuraj* et al. / INTUITIONISTIC Q-FUZZY HX GROUP/ IJMA- 2(7), July-2011, Page: 1133-1139 5.4 Theorem: is an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ. The sequence of level sub HX groups
{ < n, n > / n∈N} is a nested family if n is decreasing and n is an increasing, where N={0,1,2,3,…}.
Proof: The level sub HX groups of an intuitionistic Q-fuzzy HX subgroup of a HX group ϑ form a chain. Since µ(E, q ) µ(A, q ) and ν (E, q ) ν(A, q ) for all A in ϑ and q∈ Q , therefore < 0, 0 > , where µ(E, q )= 0 and ν(E, q ) = 0 is the smallest and we have the chain :
{E}⊆ < 0, 0 > ⊆ < 1, 1 > ⊆ < 2, 2 > ⊆ … ⊆ < n, n > = ϑ, where 0 > 1 > 2 > …… > n and 0 < 1 < 2 < …… < n.
5.5 Theorem: Any sub HX group H of a HX group ϑ can be realized as a level sub HX group of some intuitionistic Q-fuzzy HX subgroup of ϑ.
Proof: Let be an intuitionistic Q-fuzzy subset and A∈ϑ and q ∈ Q. Define,
0 if A ∉ H
µ (A, q ) =
if A ∈ H , where ∈( 0,1].
0 if A ∈ H ν (A, q ) =
if A ∉ H , where ∈( 0,1].
We shall prove that is an intuitionistic Q-fuzzy HX subgroup of ϑ.
Let A, B ∈ϑ and q ∈ Q,
(i) Suppose A, B ∈ H, then AB ∈ H and AB-1∈ H.
µ (A, q ) = , µ (B, q ) = , and µ (AB-1, q ) = .
ν (A, q ) = 0 , ν (B, q ) = 0, and (AB-1, q ) = 0.
Hence µ (AB-1, q ) min { µ (A, q ) , µ(B, q ) } and ν (AB-1, q ) max { ν (A, q ) , ν (B, q ) }.
(ii) Suppose A ∈ H and B ∉ H, then AB ∉ H and AB-1∉ H.
µ (A, q ) = , µ (B, q ) = 0 and µ (AB-1, q ) = 0.
ν (A, q ) = 0, ν (B, q ) = and ν (AB-1, q ) = .
Hence µ (AB-1, q ) min { µ(A, q ) , µ(B, q ) } and ν (AB-1, q ) max { ν (A, q ) , ν (B, q ) }.
(iii) Suppose A, B ∉ H, then AB-1∈ H or AB-1∉ H.
µ (A, q ) = 0, µ (B, q ) = 0 and µ (AB-1, q ) = or 0.
ν (A, q ) = , (B, q ) = and ν (AB-1, q ) = 0 or .
Hence µ (AB-1, q ) min { µ (A, q ) , µ(B, q ) } and ν (AB-1, q ) max { ν (A, q ) , ν (B, q ) }.
Thus in all cases, is an intuitionistic Q-fuzzy HX subgroup of ϑ.
R. Muthuraj* et al. / INTUITIONISTIC Q-FUZZY HX GROUP/ IJMA- 2(7), July-2011, Page: 1133-1139
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