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INTUITIONISTIC S-FUZZY SOFT NORMAL SUBGROUPS
R.Nagarajan*1
*1Professor, Department Of Science& Humanities J.J College Of Engineering & Technology
Ammapettai. Poolankulathupatti, Tiruchirappalli-620012 Tamilnadu, India.
ABSTRACT
In this paper, we study the cartesian product of intuitionistic fuzzy soft normal subgroup structure over snorm.
By using s-norm of S, we characterize some basic results of intuitionistic S-fuzzy soft subgroup of normal
subgroup. Also, we define the relationship between intuitionistic S-fuzzy soft subgroup and intuitionistic S-
fuzzy soft normal subgroup. Finally we prove some basic properties.
Keywords: Arbitrary Group , S-Norm, Idempotent, Soft Set, Intuitionistic Fuzzy Soft Subgroup,Additive Group,
Relation, Bounded Sum, Product.
I. INTRODUCTION
Soft set is a parameterized general mathematical tool which deals with a collection of approximate
descriptions of objects. Each approximate description has two parts, a predicate and an approximate value set.
In classical mathematics, a mathematical model of an object is constructed and defines the notion of exact
solution of this model. Usually the mathematical model is too complicated and the exact solution is not easily
obtained. So, the notion of approximate solution is introduced and the solution is calculated. The origin of Soft
Set theory could be traced from the work of Pawlak [12] in 1993 entilted hard and soft set in proceeding of the
International E Workshop on rough sets and discovery at Banff. His notion of soft sets is a unified view of
classical, rough and fuzzy sets. In order to solidify the theory of soft Set, P.K. Maji et al., [10] in 2002, defined
some basic terms of the theory such as equality of two soft sets, subsets and super set of a soft set, complement
of a soft set, null soft set, and absolute soft set with examples. Binary operations like AND, OR, union and
intersection were also defined. This motivated D. Molodtsov’s work [11] in 1999 titled soft set theory first
results. Therein, the basic notions of the theory of soft sets and some of its possible applications were
presented. For positive motivation, the work discusses some problems of the future with regard to the theory.
The notion of a fuzzy subset of a set is due to Lotfi Zadeh [13]. At present this concept has been applied to many
mathematical branches, such as group, functional analysis, probability theory, topology, and so on. The notion
of fuzzy subgroup was introduced by A. Rosenfeld et.al [[5], [10]] in his pioneering paper. Many authors [[2],
[3], [6], [7], [8]] applied the concept of fuzzy sets for studies in fuzzy semi groups, fuzzy groups, fuzzy rings,
fuzzy ideals, fuzzy semi rings and fuzzy near-rings and so on. In fact many basic properties in group theory are
found to be carried over to fuzzy groups. In 1979 Anthony and Sherwood [4] redefined a fuzzy subgroup of a
group using the concept of triangular norm (t-norm, for short). In this paper, we study the cartesian product of
intuitionistic fuzzy soft normal subgroup structure over s-norm. By using s-norm of S, we characterize some
basic results of intuitionistic S-fuzzy soft subgroups of normal subgroups. Also,we define the relationship
between intuitionistic S-fuzzy soft subgroup and intuitionistic S-fuzzy soft normal subgroup Finally we prove
some basic properties.
II. PRELIMINARIES
Definition-2.1: Let G1 and G2 be two arbitrary groups with a multiplication binary operations and identities e1,
e2 respectively. A fuzzy subset of G1 x G2 , we mean function from G1 x G2 into [ 0,1]. The set of all fuzzy subsets
of G1 x G2 is called the [0,1] –power set of G1 x G2 and is denoted by [ 0,1] G1xG2.
Defintion-2.2: By an s-norm S, we mean a function S : [0,1] x [0,1]  [0,1] satisfying the following conditions;
(S1) S(x, 0) = x
(S2) S(x ,x) ≤ S(y, z) if y ≤ z.
(S3) S(x, y) = S(y, x)
(S4) S (x , S(y, z)) = S (S(x, y), z), for all x, y, z ε [ 0,1].
Proposition-2.3: For an s-norm S, then the following statements holds S(x, y) ≥ max {x ,y}, for all x, y ε [ 0,1].
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we say that S is idempotent if for all x ε [ 0,1], S( x, x ) = x.
Example-2.4: The basic s-norms are
Sm(x, y) = max {x, y}
Sb(x, y) = min {0, x+y -1} and
Sp(x, y) = xy, which are called standard union, bounded sum, and algebraic product respectively.
Definition-2.5: A soft set over U is defined as : E P(U) such that = if A.
In other words, a soft set U is a parameterized family of subsets of the universe U. For all
A may be considered as the set of -approximate elements of the soft set . A soft set over U can be
presented by the set of ordered pairs:
= {( ) E, =P (U)}……….(1)
Clearly, a soft set is not a set. For illustration, Molodtsov consider several examples in [11].
If is a soft set over U, then the image of is defined by Im( ) = { }. The set of all soft sets
over U will be denoted by S(U). Some of the operations of soft sets are listed as follows.
Definition 2.6: Let S(U). If , for all E, then is called a soft subset of and denoted by
. are called soft equal, denoted by if and only if and
Definition 2.7: An intuitionistic fuzzy set (IFS) in the universe of discourse X is characterized by two
membership functions given by ₯ = { (tA(x), fA(x)) / x ε X} such that tA(x) + fA(x) ≤ 1, for all x ε X.
Definition 2.8: Let ₯ be an intuitionistic fuzzy soft subset of a group G1xG2.Then ₯ is called a intuitionistic
fuzzy soft subgroup of G1xG2 under a s-norm S( INT S-fuzzy soft subgroup) iff for all (a, b), (c, d) ε G1xG2
(INT- SFSG1) ₯((a, b) (c, d) ) ≤ S(₯(a, b), ₯(c, d))
(INT -SFSG2) ₯(a, b)-1 ≤ ₯(a, b).
Denote by INT -SFS (G1xG2), the set of all INT S-fuzzy soft subgroups of G1xG2.
Example 2.9: Let Z2 = {0,1}, Z3 = {0, 1,2} be two additive groups. Then
Z1xZ2 = { (0,0), (0,1), (0,2),(1,0), (1,1), (1,2)}. Define intuitionistic fuzzy soft set ₯ in Z2 x Z3 by ₯ (0,0) = 0.4, ₯
(1,0) = 0.6, ₯ (0,2) = ₯(0,1) = 0.7, ₯ (1,1) = ₯(1,2) = 0.8. If S(x,y) = Sb(x,y) = min {0, x +y-1 }, for all (x,y) ε
Z2xZ3 , then A ε SFS (Z1x Z2).
Definition 2.10 : Let ₯1,₯2εINT-SFS(G1xG2) and (a,b) ε G1xG2. We define
(i) ₯1 ₯2 iff ₯1(a, b) ≥ ₯2 (a, b),
(ii) ₯1 =₯.2 iff ₯1(a, b) =₯2(a, b),
(iii) (₯1 ₯2) (a ,b) = S {₯1 (a, b) , ₯2 (a, b)}.
Also ₯1 ₯2 = ₯2 ₯1 and ₯1 (₯2 ₯3) = (₯1 ₯2) ₯3 = ₯1 (₯2 ₯3) (property( S3 and S4)).
Lemma 2.11: Let S be a s-norm. Then S (S(a, b), S(w, c)) = S(S(a, w), S(b, c)) for all a, b, w, c ε [ 0,1].
III. PROPERTIES OF INTUITIONISTIC S-FUZZY SOFT SUBGROUP STRCTURES
Proposition 3.1: Let ₯1, ₯2 ε INT-SFS (G1 xG2). Then ₯1 ₯2 ε INT-SFS (G1 xG2).
Proof: Let (a ,b), (c ,d) ε G1 x G2.
(₯1 ₯2 ) ( (a, b) (c, d) ) = S (₯1((a, b) (c, d)), ₯2((a, b)(c, d)))
≤ S (S (₯1(a, b), ₯1(c, d) , S(₯2(a, b), ₯2(c, d)))
= S(S (₯1(a, b), ₯2(a, b), S(₯1(c, d), ₯2(c, d)).
Also
(₯1 ₯2 ) ( (a,b)-1 = S(₯1(a ,b )-1, ₯2(a, b)-1)
≤ S(₯1(a, b), ₯2(a, b))
= (₯1 ₯2 ) ( (a, b).
Corollary 3.2: Let Jn = { 1, 2, 3 …… n }. If { ₯i / i ε Jn} INT-SFS (G1xG2). Then ₯ = ⋃ ε SFS (G1xG2).
Example 3.3: Let Z3 = {0,1,2} be an additive group. Then
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Z3 x Z3 = { (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2)}. Define intuitionistic fuzzy soft sets ₯1, ₯2 in
Z3x Z3 by
₯1(0,0) = 0.5 ₯2(0,0) = 0.3
₯1(0,1) = 0.5 ₯2(0,1) = 0.3
₯1(0,2) = 0.6 ₯2(0,2) = 0.4
₯1(1,0) = 0.6 ₯2(1,0) = 0.4
₯1(2,0) = 0.7 ₯2(2,0) = 0.5
₯1(1,1) = 0.7 ₯2(1,1) = 0.5
₯1(2,2) = 0.8 ₯2(2,2) = 0.6
₯1(2,1) = 0.8 ₯2(2,1) = 0.6
₯1(1,2) = 0.9 ₯2(1,2) = 0.7
respectively. If S(a,b) = Sb(a, b) = min { 0, a +b -1}, for all (a, b) ε Z3 xZ3, then ₯1, ₯2, ₯1 ₯2 ε INT-
SFS(Z3xZ3).
Lemma 3.4: Let ₯ be an intuitionistic fuzzy soft subset of a finite group G1 x G2 and S be idempotent. If ₯
satisfies condition (INT-SFSG1) of Definition -2.8, then ₯ ε INT-SFS (G1 xG2).
Proof: Let (a, b) ε G1xG2.
(a ,b) ≠ (e1 , e2 ).
Since G1 xG2 is finite, (a,b) has finite order, say n > 1.
So (a, b)n = (e1, e2) and (a, b)-1 = (a , b) n-1.
Now by using (INT-SFSG1) repeatedly, we have that
₯ ((a, b)-1) = ₯ (( a, b)n-1)
= ₯ ((a, b)n-2 (a, b))
≤ S(₯ (a, b)n-1 , ₯ (a, b))
≤ S(₯ (a,b), ₯ (a,b),……… ₯ (a,b)) (n-times)
= ₯ (a,b).
Lemma 3.5: Let ₯ εINT-SFS(G1 xG2 ). If S be an idempotent. Then for all (x,y) ε G 1X G2 , and n ≥ 1,
(i) ₯ (e1, e2 ) n ≤ ₯ (a,b);
(ii) ₯ (a,b)n ≤ ₯ (a,b);
(iii) ₯ (a,b) = ₯ (a,b)-1.
Proof : Let (a,b) ε G1 xG2 and n ≥ 1.
(i) ₯ (e1, e2)n = ₯ ((a,b) (a,b)-1)n
≤ S(₯ (a,b), ₯ (a,b)-1)n
≤ S(₯ (a,b), ₯ (a,b), ……….. ₯ (a,b)) (n-times)
= ₯ (a,b).
(ii) ₯ (a,b)n = ₯ ((a,b), (a,b) ……… (a,b))
≤ S (₯ (a,b), ₯ (a,b) ……….. ₯ (a,b)) (n-times)
= ₯ (a,b)
(iii) ₯ (a,b) = ₯ ((a,b)-1) ≤ ₯ (a,b)-1 ≤ ₯ (a,b).
So ₯ (a,b) = ₯ (a,b)-1.
Proposition 3.6: Let ₯ εINT-SFS (G1 xG2) and (a,b) ε G1xG2. If S be idempotent, then ₯ ((a,b)(c,d)) = ₯ (c,d)
for all (c,d) ε G1xG2 if and only if ₯ (a,b) = ₯ (e1, e2).
Proof: Suppose that ₯ ((a,b) (c,d)) = ₯ (c,d), for all (c,d) ε G1xG2.
Then by letting (c,d) = (e1,e2), we get that ₯ (a,b) = ₯ (e1, e2).
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Conversly, suppose that ₯ (a,b) = ₯ (e1,e2).
By lemma-3.5, we have
₯ (a,b) ≤ ₯ ((a,b)(c,d)), A(c,d)).
Now
₯ ((a,b)(c,d)) ≤ S (₯ (a,b), ₯ (c,d)) ≤ S(₯ (c,d), ₯ (c,d)) = ₯ (c,d).
Also
₯ (c,d) = ₯ ((a,b)-1(a,b)(c,d)))
≤ S(₯ (a,b), ₯ (a,b),(c,d)))
≤ S (₯ (a,b)(c,d), ₯ ((a,b)(c,d))
= ₯ ((a,b)(c,d)).
Example 3.7: Let Z2 = {0,1} and ₯ be a intuitionistic fuzzy soft set in Z1 xZ2 as
₯ (0,0) = 0.2, ₯ (1,0) = 0.4, ₯ (0,1) = 0.3, ₯ (1,1) = 0.5.
If S(a,b) = Sm(a,b) = max {a,b} ,for all (a,b) ε Z1xZ2, then ₯ ((a,b)(c,d)) = ₯ (c,d) for all (c,d) ε Z1xZ2 if and only if
₯ (a,b) = ₯ (0,0).
Definition 3.8: Let ɸ be a mapping from G1xG2 into H1xH2, ₯ ε [0,1] G1xG1 and α ε [0,1]H1xH2 . By (D.S Malik)
ɸ(₯) ε [0,1] H1xH2 and ɸ-1(α) ε [0,1] G1xG2 , defined by for all (c,d) ε H1x H2, ɸ(₯) (c,d) = inf { ₯ (a,b) / (a,b) ε
G1 xG2, f(a,b) = (c,d)} if ɸ-1(c,d) is empty. Also for all (a,b) ε G1x G2, ɸ-1(α) (a,b) = α (ɸ(a,b)).
Lemma 3.9: Let ₯ ε INT-SFS(G1xG2) and H1xH2 be a group. Suppose that ɸ is a homomorphism of G1xG2 into
H1xH2. Then ɸ(₯) εINT-SFS(H1xH2).
Proof: Let (α1, α2 ), (β1,β2) ε H1xH2 and (a,b),(c,d) ε G1x G2.If (α1, α2 ) ε ɸ(G1xG2) or (β1, β2 ) ε ɸ(G1xG2), then
ɸ(₯)(α1, α2) = ɸ(₯) (β1, β2) = 0
≤ ɸ(₯)((α1, α2)(β1, β2)).
Suppose (α1, α2 ) = ɸ (a,b) and (β1, β2) = ɸ (c,d), then
ɸ(₯)((α1,α2)(β1, β2)) = inf { (₯ (a,b) (c,d) / (α1, α2 )
= ɸ (a,b)and (β1, β2) = ɸ (c,d)}
≤ inf { S(₯ (a,b), ₯ (c,d) / (α1, α2 )
= ɸ (a,b)and (β1, β2) = ɸ (c,d)}
= S (inf { (₯ (a,b) / (α1, α2 )
= ɸ (a,b), inf {₯ (c,d) / (β1, β2) = ɸ (c,d)}
= S(ɸ(₯)(α1, α2), ɸ(₯)(β1, β2)),
Also since ₯ ε INT-SFS(G1xG2), we have
ɸ(₯)(α1, α2)-1) = ɸ(₯)(α1, α2).
Lemma 3.10: Let H1 x H2 be a group and α ε INT-SFS(H1xH2). If ɸ be a homomorphism of G1xG2 into H1xH2, then
ɸ-1(α) εINT-SFS(G1xG2).
Proof: Let (a,b), (c,d) ε G1xG2.
Then ɸ-1(α) ((a,b)(c,d)) = α (ɸ(a,b)(c,d))
= α (ɸ(a, b) ɸ(c,d))
≤ S(α(ɸ(a,b), α(ɸ(c,d))
= S(ɸ-1(α)(a,b) , ɸ-1(α)(c,d)).
The proof is completed.
IV. INTUITIONISTIC S-FUZZY SOFT NORMAL SUBGROUP
Definition 4.1: we say that ₯ εINT-SFS(G1xG2) is a normal S-fuzzy soft subgroup of G1xG2 if for all (a,b), (c,d) ε
G1xG2, ₯ ((a,b)(c,d)(a,b)-1) = ₯ (c,d).Also we denote by INT-NSFS(G1xG2) the set of intuitionistic S-fuzzy
normal soft subgroups of G1xG2.
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Proposition 4.2: Let ₯ εINT-NSFS(G1xG2) and H1xH2 be a group. Suppose that ɸ is an epimorphism of G1xG2
onto H1xH2. Then ɸ(₯) εINT-NSFS(H1xH2).
Proof: From Lemma-3.9, we have ɸ(₯) ε INT-SFS(H1xH2).
Let (a,b),(c,d) ε H1xH2.
Since f is a surjective function ,ɸ(α1,α2) = (a,b) for some (α1, α2 )ε G1xG2.
Then
ɸ(₯)((a,b)(c,d)(a,b)-1) = inf { ₯ (w1,w2) / (w1,w2) ε G1xG2, ɸ(w1,w2) = (a,b)(c,d)(a,b)-1}
= inf { ₯ (α1,α2)-1(w1,w2) (α1,α2) / (w1,w2) ε G1xG2, ɸ(α1,α2)-1(w1,w2)(α1,α2) = (c,d)}
= inf { ₯ (w1,w2) / (w1,w2) ε G1xG2, ɸ(w1,w2) =(c,d) }
= ɸ(₯)(c,d).
Hence the proof.
Proposition 4.3: Let H1xH2 be a group and α ε INT-NSFS(G1xG2). Suppose that ɸ is a homomorphism of G1xG2
into H1xH2.Then ɸ-1(α) ε INT-NSFS(G1xG2).
Proof: By lemma-3.9,
ɸ-1(α) ε INT-SFS(G1xG2). Now for any (a,b),(c,d) ε G1xG2,
we have ɸ-1(α) ((a,b)(c,d)(a,b)-1) = α (ɸ((a,b)(c,d)(a,b)-1))
= α (ɸ(a,b) ɸ(c,d) ɸ(a,b)-1)
= α (ɸ(a,b) ɸ(c,d) ɸ-1(a,b))
= α (ɸ(c,d)) = ɸ-1(α)(c,d).
Hence ɸ-1(α) ε INT-NSFS(G1xG2).
Example 4.4: Let ɸ be an epimorphism of Z2x Z2 onto Z2xZ2 such that
f(a,b) = (a,b) for all (a,b) ε Z2xZ2.
Let S(a,b) = Sp(a,b) = ab for all (a,b) ε Z2xZ2 and define ₯ ε INT-NSFS(Z2xZ2) as
₯ (0,0) = ₯ (0,1) = ₯ (1,0) = ₯ (1,1) = 0.7. Then ɸ(₯), ɸ-1(₯) ε INT-NSFS(Z2xZ2).
Proposition-4.5: Let ₯1, ₯2ε INT-NSFS(Z2xZ2). Then ₯1 ₯2ε INT-NSFS(Z2xZ2).
Proof: Since ₯1, ₯2ε INT-NSFS(Z2xZ2), then from definition-4.1, we have that
(₯1 ₯2)((a,b)(c,d)(a,b)-1) = S(₯1(a,b)(c,d)(a,b)-1, ₯1(a,b)(c,d)(a,b)-1)
= S(₯1(c,d) , ₯2(c,d))
= (₯1 ₯2)(c,d).
Corollary 4.6: Let Jn = {1,2,3…..n }. If {₯i / i ε Jn} is a subset of INT-NSFS(Z2xZ2).Then ₯ = ⋃ εINT-
NSFS(G1xG2).
Example 4.7: Let ₯1(0,0) = ₯1(0,1) = ₯1(0,2) = ₯1(1,0) = ₯1(1,1) = ₯1(1,2) = 0.2 and
₯2(0,0) = ₯2(0,1) = ₯2(0,2) = ₯2(1,0) = ₯2(1,1) = ₯2(1,2) = 0.1.
Let S(a,b) = Sp(a,b) = ab for all (a,b) ε Z2xZ2. Then ₯1, ₯2, ₯1 ₯2ε INT-NSFS(Z2xZ2).
Definition 4.8: Let ₯ ε INT-SFS(G1xG2) and ₯ is subset of α. Then ₯ is called an intuitionistic fuzzy soft
normal subgroup of the subgroup α, written ₯ α if for all (a,b),(c,d) ε G1xG2, ₯ ((a,b)(c,d)(a,b)-1) ≤ S(₯ (c,d),
α(a,b)).
Example 4.9: Let G = { 1,-1} be a productive group and ₯ (-1,-1) = ₯ (1,1) = 0.5, ₯ (1,-1) = ₯ (-1, 1) = 0.7. If
S(a,b) = Sm(a,b) = max { a,b} for all (a,b) ε G1xG2, then ₯ α. Also ₯ ε INT-SFS(G1xG2) if and only if ₯ G1xG2.
Lemma 4.10: Let S be idempotent. If ₯ ε INT-SFS(G1xG2) and α ε INT-SFS(G1xG2), then ₯ α α.
Proof: From proposition-3.1, we have (₯ α) ε INT-SFS(G1xG2).
Now for all (a,b),(c,d) ε G1xG2.we have
(₯ α)((a,b)(c,d)(a,b)-1) = S (₯ ((a,b)(c,d)(a,b)-1, α ((a,b)(c,d)(a,b)-1)
= S(₯ (c,d), α((a,b)(c,d)(a,b)-1))
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≤ S(₯ (c,d), S(S(α(a,b),α(c,d), α(a,b)))
= S(₯ (c,d), S(α(c,d),α(a,b))
= S(S(₯ (c,d), α(c,d), α(a,b))
= S((₯ α)(c,d), α(a,b)). Hence ₯ α α.
Proposition 4.11: Let ₯ , α ε INT-SFS(G1xG2) and ₯ α. Let H1xH2 be a group and ɸ a homomorphism from
G1xG2 into H1xH2. Then ɸ(₯) ɸ(α).
Proof: By lemma-3.9, we have ɸ(₯), ɸ(α) ε SFS(H1xH2). Let (a,b), (c,d) ε H1xH2 and (u1, u2), (α1,α2) ε G1xG2.
Then
ɸ(₯)((a,b)(c,d)(a,b)-1))
= inf { ₯ (z) / z ε G1xG2, ɸ(z) = (a,b)(c,d)(a,b)-1}
= inf { ₯ (u1,u2)(v1,v2) (u1,u2)-1) / (u1,u2), (α1,α2) ε G1xG2, ɸ(u1,u2)= (a, b), ɸ(α1,α2) = (c,d)}
= inf { S(₯ (α1,α2) ,α (u1,u2) / ɸ(u1,u2) =(a,b), ɸ(α1, α2) = (c,d) }
= S(inf {₯ (α1,α2)/ (c,d) = ɸ (α1, α2)}, (inf {α(u1,u2)/ (a,b) = ɸ (u1, u2)}
= S (ɸ(₯)(c , d), ɸ(₯)(a,b)).
Hence ɸ(₯) ɸ(α).
Proposition-4.12: Let H1xH2 be a group. Let ₯ , α ε INT-SFS(H1xH2) and A α. If ɸ be a homomorphism from
G1xG2 into H1xH2, then ɸ-1(A) ɸ-1(α).
Proof: From Lemma-3.10, we have ɸ-1(₯), ɸ-1(α) ε INT-SFS(G1xG2) .
Let (a,b),(c,d) ε G1xG2. Now
ɸ-1(₯)((a,b)(c,d)(a,b)-1) = ₯ (ɸ(a,b)(c,d)(a,b)-1)
= ₯ (ɸ(a,b) ɸ (c,d) ɸ-1(a,b))
≤ S (₯ (ɸ(y), α(ɸ(a,b))
= S (ɸ-1(₯)(c,d), ɸ-1(α)(a,b)), hence ɸ-1(₯) ɸ-1(α).
Example 4.13: Let ɸ be a homomorphism from Z2x Z2 into Z2xZ2 such that ɸ(a,b) = (a,b)-1 for all (a,b) ε Z2xZ2.
Let S(a,b) = Sp(a,b) = ab for all (a,b) ε Z2xZ2. Define ₯, α ε INT-SFS(H1xH2) as ₯ (0,0) = 0.3, ₯ (0,1) = 0.4, ₯
(1,0)= 0.5, ₯ (1,1) = 0.6, α(0,0) = 0.5, α(0,1) = 0.5,α(1,0) = 0.5,α(1,1) = 0.5. Now we have that ₯ α. ɸ(₯)
ɸ(α), ɸ-1(₯) ɸ-1(α).
V. CONCLUSION
By using s-norm of S, we characterize some basic properties of INT S-fuzzy soft subgroups and cartesian
product of normal subgroups. In Classical mathematics, a Mathematical Model of an object is constructed and
defines the notion of exact solution of this model. Usually the mathematical model is too complicated and the
exact solution is not easily obtained. So, the notion of approximate solution is introduced and the solutionis
calculated Also ,we define the relational concept of INT S-fuzzy soft subgroup , INT S-fuzzy normal soft
subgroup and prove some basic properties.
VI. REFERENCES
[1] M. T. Abu Osman, On some products of fuzzy subgroups, Fuzzy Sets and Systems, 24 (1987) 79-86.
https://doi.org/10.1016/0165-0114(87)90115-1
[2] J. Ahsan, R. M. Latif, M. Shabir, Fuzzy quasi -ideals in semigroups, The J. Fuzzy Math, 9 (2) (2001) 259-
270.
[3] M. I. Ali, M. Shabir, Soft ideals and generalized fuzzy ideals in semigroups, New Math. Nat. Comp, 5
(2009) 599-615. https://doi.org/10.1142/S1793005709001544
[4] J. M. Anthony, H. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl, 69 (1979) 124-130.
https://doi.org/10.1016/0022-247X(79)90182-3
[5] K. R. Bhutani, J. N. Mordeson, A. Rosenfeld, Fuzzy Group Theory, Springer-Verlag Berlin Heidelberg, 182
(2005) 1-39. https://doi.org/10.1007/b12359
e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com
www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
[719]
[6] Y. Cheng-Fu, Fuzzy soft semigroups and fuzzy soft ideals, Comput. Math. Appl, 61 (2011) 255-561.
https://doi.org/10.1016/j.camwa.2010.10.047
[7] F. Feng, Y. B. Jun, X. Z. Zhao, Soft semirings, Comput. Math. Appl, 56 (2005) 2621-2628.
https://doi.org/10.1016/j.camwa.2008.05.011
[8] N. Kuroki, D. S. Malik, J. N. Mordeson, Fuzzy semigroups, Springer, (2003).
https://doi.org/10.1007/978-3-540-37125-0
[9] D. S. Malik, J. N. Mordeson, Fuzzy Commutative Algebra, World Science publishing Co.Pte.Ltd, (1995).
[10] A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl, 35 (1971) 512-517. https://doi.org/10.1016/0022-
247X(71)90199-5
[11] Maji, P.K., Biswas, R. and Roy, A.R. (2003). Soft set theory. Computers and Mathematics with
Applications, 45: 555-562.
[12] Molodtsov, D. (1999). Soft set Theory-First results. Computers and Mathematics with Applications, 37:
19-31.
[13] Pawlak, Z. (1982). Rough sets. International Journal of Information and Computer 134 Sciences, 11:
341-356.
[14] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965) 338-353. https://doi.org/10.1016/S0019-
9958(65)90241-X.

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INTUITIONISTIC S-FUZZY SOFT NORMAL SUBGROUPS

  • 1. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [713] INTUITIONISTIC S-FUZZY SOFT NORMAL SUBGROUPS R.Nagarajan*1 *1Professor, Department Of Science& Humanities J.J College Of Engineering & Technology Ammapettai. Poolankulathupatti, Tiruchirappalli-620012 Tamilnadu, India. ABSTRACT In this paper, we study the cartesian product of intuitionistic fuzzy soft normal subgroup structure over snorm. By using s-norm of S, we characterize some basic results of intuitionistic S-fuzzy soft subgroup of normal subgroup. Also, we define the relationship between intuitionistic S-fuzzy soft subgroup and intuitionistic S- fuzzy soft normal subgroup. Finally we prove some basic properties. Keywords: Arbitrary Group , S-Norm, Idempotent, Soft Set, Intuitionistic Fuzzy Soft Subgroup,Additive Group, Relation, Bounded Sum, Product. I. INTRODUCTION Soft set is a parameterized general mathematical tool which deals with a collection of approximate descriptions of objects. Each approximate description has two parts, a predicate and an approximate value set. In classical mathematics, a mathematical model of an object is constructed and defines the notion of exact solution of this model. Usually the mathematical model is too complicated and the exact solution is not easily obtained. So, the notion of approximate solution is introduced and the solution is calculated. The origin of Soft Set theory could be traced from the work of Pawlak [12] in 1993 entilted hard and soft set in proceeding of the International E Workshop on rough sets and discovery at Banff. His notion of soft sets is a unified view of classical, rough and fuzzy sets. In order to solidify the theory of soft Set, P.K. Maji et al., [10] in 2002, defined some basic terms of the theory such as equality of two soft sets, subsets and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Binary operations like AND, OR, union and intersection were also defined. This motivated D. Molodtsov’s work [11] in 1999 titled soft set theory first results. Therein, the basic notions of the theory of soft sets and some of its possible applications were presented. For positive motivation, the work discusses some problems of the future with regard to the theory. The notion of a fuzzy subset of a set is due to Lotfi Zadeh [13]. At present this concept has been applied to many mathematical branches, such as group, functional analysis, probability theory, topology, and so on. The notion of fuzzy subgroup was introduced by A. Rosenfeld et.al [[5], [10]] in his pioneering paper. Many authors [[2], [3], [6], [7], [8]] applied the concept of fuzzy sets for studies in fuzzy semi groups, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semi rings and fuzzy near-rings and so on. In fact many basic properties in group theory are found to be carried over to fuzzy groups. In 1979 Anthony and Sherwood [4] redefined a fuzzy subgroup of a group using the concept of triangular norm (t-norm, for short). In this paper, we study the cartesian product of intuitionistic fuzzy soft normal subgroup structure over s-norm. By using s-norm of S, we characterize some basic results of intuitionistic S-fuzzy soft subgroups of normal subgroups. Also,we define the relationship between intuitionistic S-fuzzy soft subgroup and intuitionistic S-fuzzy soft normal subgroup Finally we prove some basic properties. II. PRELIMINARIES Definition-2.1: Let G1 and G2 be two arbitrary groups with a multiplication binary operations and identities e1, e2 respectively. A fuzzy subset of G1 x G2 , we mean function from G1 x G2 into [ 0,1]. The set of all fuzzy subsets of G1 x G2 is called the [0,1] –power set of G1 x G2 and is denoted by [ 0,1] G1xG2. Defintion-2.2: By an s-norm S, we mean a function S : [0,1] x [0,1]  [0,1] satisfying the following conditions; (S1) S(x, 0) = x (S2) S(x ,x) ≤ S(y, z) if y ≤ z. (S3) S(x, y) = S(y, x) (S4) S (x , S(y, z)) = S (S(x, y), z), for all x, y, z ε [ 0,1]. Proposition-2.3: For an s-norm S, then the following statements holds S(x, y) ≥ max {x ,y}, for all x, y ε [ 0,1].
  • 2. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [714] we say that S is idempotent if for all x ε [ 0,1], S( x, x ) = x. Example-2.4: The basic s-norms are Sm(x, y) = max {x, y} Sb(x, y) = min {0, x+y -1} and Sp(x, y) = xy, which are called standard union, bounded sum, and algebraic product respectively. Definition-2.5: A soft set over U is defined as : E P(U) such that = if A. In other words, a soft set U is a parameterized family of subsets of the universe U. For all A may be considered as the set of -approximate elements of the soft set . A soft set over U can be presented by the set of ordered pairs: = {( ) E, =P (U)}……….(1) Clearly, a soft set is not a set. For illustration, Molodtsov consider several examples in [11]. If is a soft set over U, then the image of is defined by Im( ) = { }. The set of all soft sets over U will be denoted by S(U). Some of the operations of soft sets are listed as follows. Definition 2.6: Let S(U). If , for all E, then is called a soft subset of and denoted by . are called soft equal, denoted by if and only if and Definition 2.7: An intuitionistic fuzzy set (IFS) in the universe of discourse X is characterized by two membership functions given by ₯ = { (tA(x), fA(x)) / x ε X} such that tA(x) + fA(x) ≤ 1, for all x ε X. Definition 2.8: Let ₯ be an intuitionistic fuzzy soft subset of a group G1xG2.Then ₯ is called a intuitionistic fuzzy soft subgroup of G1xG2 under a s-norm S( INT S-fuzzy soft subgroup) iff for all (a, b), (c, d) ε G1xG2 (INT- SFSG1) ₯((a, b) (c, d) ) ≤ S(₯(a, b), ₯(c, d)) (INT -SFSG2) ₯(a, b)-1 ≤ ₯(a, b). Denote by INT -SFS (G1xG2), the set of all INT S-fuzzy soft subgroups of G1xG2. Example 2.9: Let Z2 = {0,1}, Z3 = {0, 1,2} be two additive groups. Then Z1xZ2 = { (0,0), (0,1), (0,2),(1,0), (1,1), (1,2)}. Define intuitionistic fuzzy soft set ₯ in Z2 x Z3 by ₯ (0,0) = 0.4, ₯ (1,0) = 0.6, ₯ (0,2) = ₯(0,1) = 0.7, ₯ (1,1) = ₯(1,2) = 0.8. If S(x,y) = Sb(x,y) = min {0, x +y-1 }, for all (x,y) ε Z2xZ3 , then A ε SFS (Z1x Z2). Definition 2.10 : Let ₯1,₯2εINT-SFS(G1xG2) and (a,b) ε G1xG2. We define (i) ₯1 ₯2 iff ₯1(a, b) ≥ ₯2 (a, b), (ii) ₯1 =₯.2 iff ₯1(a, b) =₯2(a, b), (iii) (₯1 ₯2) (a ,b) = S {₯1 (a, b) , ₯2 (a, b)}. Also ₯1 ₯2 = ₯2 ₯1 and ₯1 (₯2 ₯3) = (₯1 ₯2) ₯3 = ₯1 (₯2 ₯3) (property( S3 and S4)). Lemma 2.11: Let S be a s-norm. Then S (S(a, b), S(w, c)) = S(S(a, w), S(b, c)) for all a, b, w, c ε [ 0,1]. III. PROPERTIES OF INTUITIONISTIC S-FUZZY SOFT SUBGROUP STRCTURES Proposition 3.1: Let ₯1, ₯2 ε INT-SFS (G1 xG2). Then ₯1 ₯2 ε INT-SFS (G1 xG2). Proof: Let (a ,b), (c ,d) ε G1 x G2. (₯1 ₯2 ) ( (a, b) (c, d) ) = S (₯1((a, b) (c, d)), ₯2((a, b)(c, d))) ≤ S (S (₯1(a, b), ₯1(c, d) , S(₯2(a, b), ₯2(c, d))) = S(S (₯1(a, b), ₯2(a, b), S(₯1(c, d), ₯2(c, d)). Also (₯1 ₯2 ) ( (a,b)-1 = S(₯1(a ,b )-1, ₯2(a, b)-1) ≤ S(₯1(a, b), ₯2(a, b)) = (₯1 ₯2 ) ( (a, b). Corollary 3.2: Let Jn = { 1, 2, 3 …… n }. If { ₯i / i ε Jn} INT-SFS (G1xG2). Then ₯ = ⋃ ε SFS (G1xG2). Example 3.3: Let Z3 = {0,1,2} be an additive group. Then
  • 3. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [715] Z3 x Z3 = { (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2)}. Define intuitionistic fuzzy soft sets ₯1, ₯2 in Z3x Z3 by ₯1(0,0) = 0.5 ₯2(0,0) = 0.3 ₯1(0,1) = 0.5 ₯2(0,1) = 0.3 ₯1(0,2) = 0.6 ₯2(0,2) = 0.4 ₯1(1,0) = 0.6 ₯2(1,0) = 0.4 ₯1(2,0) = 0.7 ₯2(2,0) = 0.5 ₯1(1,1) = 0.7 ₯2(1,1) = 0.5 ₯1(2,2) = 0.8 ₯2(2,2) = 0.6 ₯1(2,1) = 0.8 ₯2(2,1) = 0.6 ₯1(1,2) = 0.9 ₯2(1,2) = 0.7 respectively. If S(a,b) = Sb(a, b) = min { 0, a +b -1}, for all (a, b) ε Z3 xZ3, then ₯1, ₯2, ₯1 ₯2 ε INT- SFS(Z3xZ3). Lemma 3.4: Let ₯ be an intuitionistic fuzzy soft subset of a finite group G1 x G2 and S be idempotent. If ₯ satisfies condition (INT-SFSG1) of Definition -2.8, then ₯ ε INT-SFS (G1 xG2). Proof: Let (a, b) ε G1xG2. (a ,b) ≠ (e1 , e2 ). Since G1 xG2 is finite, (a,b) has finite order, say n > 1. So (a, b)n = (e1, e2) and (a, b)-1 = (a , b) n-1. Now by using (INT-SFSG1) repeatedly, we have that ₯ ((a, b)-1) = ₯ (( a, b)n-1) = ₯ ((a, b)n-2 (a, b)) ≤ S(₯ (a, b)n-1 , ₯ (a, b)) ≤ S(₯ (a,b), ₯ (a,b),……… ₯ (a,b)) (n-times) = ₯ (a,b). Lemma 3.5: Let ₯ εINT-SFS(G1 xG2 ). If S be an idempotent. Then for all (x,y) ε G 1X G2 , and n ≥ 1, (i) ₯ (e1, e2 ) n ≤ ₯ (a,b); (ii) ₯ (a,b)n ≤ ₯ (a,b); (iii) ₯ (a,b) = ₯ (a,b)-1. Proof : Let (a,b) ε G1 xG2 and n ≥ 1. (i) ₯ (e1, e2)n = ₯ ((a,b) (a,b)-1)n ≤ S(₯ (a,b), ₯ (a,b)-1)n ≤ S(₯ (a,b), ₯ (a,b), ……….. ₯ (a,b)) (n-times) = ₯ (a,b). (ii) ₯ (a,b)n = ₯ ((a,b), (a,b) ……… (a,b)) ≤ S (₯ (a,b), ₯ (a,b) ……….. ₯ (a,b)) (n-times) = ₯ (a,b) (iii) ₯ (a,b) = ₯ ((a,b)-1) ≤ ₯ (a,b)-1 ≤ ₯ (a,b). So ₯ (a,b) = ₯ (a,b)-1. Proposition 3.6: Let ₯ εINT-SFS (G1 xG2) and (a,b) ε G1xG2. If S be idempotent, then ₯ ((a,b)(c,d)) = ₯ (c,d) for all (c,d) ε G1xG2 if and only if ₯ (a,b) = ₯ (e1, e2). Proof: Suppose that ₯ ((a,b) (c,d)) = ₯ (c,d), for all (c,d) ε G1xG2. Then by letting (c,d) = (e1,e2), we get that ₯ (a,b) = ₯ (e1, e2).
  • 4. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [716] Conversly, suppose that ₯ (a,b) = ₯ (e1,e2). By lemma-3.5, we have ₯ (a,b) ≤ ₯ ((a,b)(c,d)), A(c,d)). Now ₯ ((a,b)(c,d)) ≤ S (₯ (a,b), ₯ (c,d)) ≤ S(₯ (c,d), ₯ (c,d)) = ₯ (c,d). Also ₯ (c,d) = ₯ ((a,b)-1(a,b)(c,d))) ≤ S(₯ (a,b), ₯ (a,b),(c,d))) ≤ S (₯ (a,b)(c,d), ₯ ((a,b)(c,d)) = ₯ ((a,b)(c,d)). Example 3.7: Let Z2 = {0,1} and ₯ be a intuitionistic fuzzy soft set in Z1 xZ2 as ₯ (0,0) = 0.2, ₯ (1,0) = 0.4, ₯ (0,1) = 0.3, ₯ (1,1) = 0.5. If S(a,b) = Sm(a,b) = max {a,b} ,for all (a,b) ε Z1xZ2, then ₯ ((a,b)(c,d)) = ₯ (c,d) for all (c,d) ε Z1xZ2 if and only if ₯ (a,b) = ₯ (0,0). Definition 3.8: Let ɸ be a mapping from G1xG2 into H1xH2, ₯ ε [0,1] G1xG1 and α ε [0,1]H1xH2 . By (D.S Malik) ɸ(₯) ε [0,1] H1xH2 and ɸ-1(α) ε [0,1] G1xG2 , defined by for all (c,d) ε H1x H2, ɸ(₯) (c,d) = inf { ₯ (a,b) / (a,b) ε G1 xG2, f(a,b) = (c,d)} if ɸ-1(c,d) is empty. Also for all (a,b) ε G1x G2, ɸ-1(α) (a,b) = α (ɸ(a,b)). Lemma 3.9: Let ₯ ε INT-SFS(G1xG2) and H1xH2 be a group. Suppose that ɸ is a homomorphism of G1xG2 into H1xH2. Then ɸ(₯) εINT-SFS(H1xH2). Proof: Let (α1, α2 ), (β1,β2) ε H1xH2 and (a,b),(c,d) ε G1x G2.If (α1, α2 ) ε ɸ(G1xG2) or (β1, β2 ) ε ɸ(G1xG2), then ɸ(₯)(α1, α2) = ɸ(₯) (β1, β2) = 0 ≤ ɸ(₯)((α1, α2)(β1, β2)). Suppose (α1, α2 ) = ɸ (a,b) and (β1, β2) = ɸ (c,d), then ɸ(₯)((α1,α2)(β1, β2)) = inf { (₯ (a,b) (c,d) / (α1, α2 ) = ɸ (a,b)and (β1, β2) = ɸ (c,d)} ≤ inf { S(₯ (a,b), ₯ (c,d) / (α1, α2 ) = ɸ (a,b)and (β1, β2) = ɸ (c,d)} = S (inf { (₯ (a,b) / (α1, α2 ) = ɸ (a,b), inf {₯ (c,d) / (β1, β2) = ɸ (c,d)} = S(ɸ(₯)(α1, α2), ɸ(₯)(β1, β2)), Also since ₯ ε INT-SFS(G1xG2), we have ɸ(₯)(α1, α2)-1) = ɸ(₯)(α1, α2). Lemma 3.10: Let H1 x H2 be a group and α ε INT-SFS(H1xH2). If ɸ be a homomorphism of G1xG2 into H1xH2, then ɸ-1(α) εINT-SFS(G1xG2). Proof: Let (a,b), (c,d) ε G1xG2. Then ɸ-1(α) ((a,b)(c,d)) = α (ɸ(a,b)(c,d)) = α (ɸ(a, b) ɸ(c,d)) ≤ S(α(ɸ(a,b), α(ɸ(c,d)) = S(ɸ-1(α)(a,b) , ɸ-1(α)(c,d)). The proof is completed. IV. INTUITIONISTIC S-FUZZY SOFT NORMAL SUBGROUP Definition 4.1: we say that ₯ εINT-SFS(G1xG2) is a normal S-fuzzy soft subgroup of G1xG2 if for all (a,b), (c,d) ε G1xG2, ₯ ((a,b)(c,d)(a,b)-1) = ₯ (c,d).Also we denote by INT-NSFS(G1xG2) the set of intuitionistic S-fuzzy normal soft subgroups of G1xG2.
  • 5. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [717] Proposition 4.2: Let ₯ εINT-NSFS(G1xG2) and H1xH2 be a group. Suppose that ɸ is an epimorphism of G1xG2 onto H1xH2. Then ɸ(₯) εINT-NSFS(H1xH2). Proof: From Lemma-3.9, we have ɸ(₯) ε INT-SFS(H1xH2). Let (a,b),(c,d) ε H1xH2. Since f is a surjective function ,ɸ(α1,α2) = (a,b) for some (α1, α2 )ε G1xG2. Then ɸ(₯)((a,b)(c,d)(a,b)-1) = inf { ₯ (w1,w2) / (w1,w2) ε G1xG2, ɸ(w1,w2) = (a,b)(c,d)(a,b)-1} = inf { ₯ (α1,α2)-1(w1,w2) (α1,α2) / (w1,w2) ε G1xG2, ɸ(α1,α2)-1(w1,w2)(α1,α2) = (c,d)} = inf { ₯ (w1,w2) / (w1,w2) ε G1xG2, ɸ(w1,w2) =(c,d) } = ɸ(₯)(c,d). Hence the proof. Proposition 4.3: Let H1xH2 be a group and α ε INT-NSFS(G1xG2). Suppose that ɸ is a homomorphism of G1xG2 into H1xH2.Then ɸ-1(α) ε INT-NSFS(G1xG2). Proof: By lemma-3.9, ɸ-1(α) ε INT-SFS(G1xG2). Now for any (a,b),(c,d) ε G1xG2, we have ɸ-1(α) ((a,b)(c,d)(a,b)-1) = α (ɸ((a,b)(c,d)(a,b)-1)) = α (ɸ(a,b) ɸ(c,d) ɸ(a,b)-1) = α (ɸ(a,b) ɸ(c,d) ɸ-1(a,b)) = α (ɸ(c,d)) = ɸ-1(α)(c,d). Hence ɸ-1(α) ε INT-NSFS(G1xG2). Example 4.4: Let ɸ be an epimorphism of Z2x Z2 onto Z2xZ2 such that f(a,b) = (a,b) for all (a,b) ε Z2xZ2. Let S(a,b) = Sp(a,b) = ab for all (a,b) ε Z2xZ2 and define ₯ ε INT-NSFS(Z2xZ2) as ₯ (0,0) = ₯ (0,1) = ₯ (1,0) = ₯ (1,1) = 0.7. Then ɸ(₯), ɸ-1(₯) ε INT-NSFS(Z2xZ2). Proposition-4.5: Let ₯1, ₯2ε INT-NSFS(Z2xZ2). Then ₯1 ₯2ε INT-NSFS(Z2xZ2). Proof: Since ₯1, ₯2ε INT-NSFS(Z2xZ2), then from definition-4.1, we have that (₯1 ₯2)((a,b)(c,d)(a,b)-1) = S(₯1(a,b)(c,d)(a,b)-1, ₯1(a,b)(c,d)(a,b)-1) = S(₯1(c,d) , ₯2(c,d)) = (₯1 ₯2)(c,d). Corollary 4.6: Let Jn = {1,2,3…..n }. If {₯i / i ε Jn} is a subset of INT-NSFS(Z2xZ2).Then ₯ = ⋃ εINT- NSFS(G1xG2). Example 4.7: Let ₯1(0,0) = ₯1(0,1) = ₯1(0,2) = ₯1(1,0) = ₯1(1,1) = ₯1(1,2) = 0.2 and ₯2(0,0) = ₯2(0,1) = ₯2(0,2) = ₯2(1,0) = ₯2(1,1) = ₯2(1,2) = 0.1. Let S(a,b) = Sp(a,b) = ab for all (a,b) ε Z2xZ2. Then ₯1, ₯2, ₯1 ₯2ε INT-NSFS(Z2xZ2). Definition 4.8: Let ₯ ε INT-SFS(G1xG2) and ₯ is subset of α. Then ₯ is called an intuitionistic fuzzy soft normal subgroup of the subgroup α, written ₯ α if for all (a,b),(c,d) ε G1xG2, ₯ ((a,b)(c,d)(a,b)-1) ≤ S(₯ (c,d), α(a,b)). Example 4.9: Let G = { 1,-1} be a productive group and ₯ (-1,-1) = ₯ (1,1) = 0.5, ₯ (1,-1) = ₯ (-1, 1) = 0.7. If S(a,b) = Sm(a,b) = max { a,b} for all (a,b) ε G1xG2, then ₯ α. Also ₯ ε INT-SFS(G1xG2) if and only if ₯ G1xG2. Lemma 4.10: Let S be idempotent. If ₯ ε INT-SFS(G1xG2) and α ε INT-SFS(G1xG2), then ₯ α α. Proof: From proposition-3.1, we have (₯ α) ε INT-SFS(G1xG2). Now for all (a,b),(c,d) ε G1xG2.we have (₯ α)((a,b)(c,d)(a,b)-1) = S (₯ ((a,b)(c,d)(a,b)-1, α ((a,b)(c,d)(a,b)-1) = S(₯ (c,d), α((a,b)(c,d)(a,b)-1))
  • 6. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [718] ≤ S(₯ (c,d), S(S(α(a,b),α(c,d), α(a,b))) = S(₯ (c,d), S(α(c,d),α(a,b)) = S(S(₯ (c,d), α(c,d), α(a,b)) = S((₯ α)(c,d), α(a,b)). Hence ₯ α α. Proposition 4.11: Let ₯ , α ε INT-SFS(G1xG2) and ₯ α. Let H1xH2 be a group and ɸ a homomorphism from G1xG2 into H1xH2. Then ɸ(₯) ɸ(α). Proof: By lemma-3.9, we have ɸ(₯), ɸ(α) ε SFS(H1xH2). Let (a,b), (c,d) ε H1xH2 and (u1, u2), (α1,α2) ε G1xG2. Then ɸ(₯)((a,b)(c,d)(a,b)-1)) = inf { ₯ (z) / z ε G1xG2, ɸ(z) = (a,b)(c,d)(a,b)-1} = inf { ₯ (u1,u2)(v1,v2) (u1,u2)-1) / (u1,u2), (α1,α2) ε G1xG2, ɸ(u1,u2)= (a, b), ɸ(α1,α2) = (c,d)} = inf { S(₯ (α1,α2) ,α (u1,u2) / ɸ(u1,u2) =(a,b), ɸ(α1, α2) = (c,d) } = S(inf {₯ (α1,α2)/ (c,d) = ɸ (α1, α2)}, (inf {α(u1,u2)/ (a,b) = ɸ (u1, u2)} = S (ɸ(₯)(c , d), ɸ(₯)(a,b)). Hence ɸ(₯) ɸ(α). Proposition-4.12: Let H1xH2 be a group. Let ₯ , α ε INT-SFS(H1xH2) and A α. If ɸ be a homomorphism from G1xG2 into H1xH2, then ɸ-1(A) ɸ-1(α). Proof: From Lemma-3.10, we have ɸ-1(₯), ɸ-1(α) ε INT-SFS(G1xG2) . Let (a,b),(c,d) ε G1xG2. Now ɸ-1(₯)((a,b)(c,d)(a,b)-1) = ₯ (ɸ(a,b)(c,d)(a,b)-1) = ₯ (ɸ(a,b) ɸ (c,d) ɸ-1(a,b)) ≤ S (₯ (ɸ(y), α(ɸ(a,b)) = S (ɸ-1(₯)(c,d), ɸ-1(α)(a,b)), hence ɸ-1(₯) ɸ-1(α). Example 4.13: Let ɸ be a homomorphism from Z2x Z2 into Z2xZ2 such that ɸ(a,b) = (a,b)-1 for all (a,b) ε Z2xZ2. Let S(a,b) = Sp(a,b) = ab for all (a,b) ε Z2xZ2. Define ₯, α ε INT-SFS(H1xH2) as ₯ (0,0) = 0.3, ₯ (0,1) = 0.4, ₯ (1,0)= 0.5, ₯ (1,1) = 0.6, α(0,0) = 0.5, α(0,1) = 0.5,α(1,0) = 0.5,α(1,1) = 0.5. Now we have that ₯ α. ɸ(₯) ɸ(α), ɸ-1(₯) ɸ-1(α). V. CONCLUSION By using s-norm of S, we characterize some basic properties of INT S-fuzzy soft subgroups and cartesian product of normal subgroups. In Classical mathematics, a Mathematical Model of an object is constructed and defines the notion of exact solution of this model. Usually the mathematical model is too complicated and the exact solution is not easily obtained. So, the notion of approximate solution is introduced and the solutionis calculated Also ,we define the relational concept of INT S-fuzzy soft subgroup , INT S-fuzzy normal soft subgroup and prove some basic properties. VI. REFERENCES [1] M. T. Abu Osman, On some products of fuzzy subgroups, Fuzzy Sets and Systems, 24 (1987) 79-86. https://doi.org/10.1016/0165-0114(87)90115-1 [2] J. Ahsan, R. M. Latif, M. Shabir, Fuzzy quasi -ideals in semigroups, The J. Fuzzy Math, 9 (2) (2001) 259- 270. [3] M. I. Ali, M. Shabir, Soft ideals and generalized fuzzy ideals in semigroups, New Math. Nat. Comp, 5 (2009) 599-615. https://doi.org/10.1142/S1793005709001544 [4] J. M. Anthony, H. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl, 69 (1979) 124-130. https://doi.org/10.1016/0022-247X(79)90182-3 [5] K. R. Bhutani, J. N. Mordeson, A. Rosenfeld, Fuzzy Group Theory, Springer-Verlag Berlin Heidelberg, 182 (2005) 1-39. https://doi.org/10.1007/b12359
  • 7. e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science ( Peer-Reviewed, Open Access, Fully Refereed International Journal ) Volume:03/Issue:08/August-2021 Impact Factor- 5.354 www.irjmets.com www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [719] [6] Y. Cheng-Fu, Fuzzy soft semigroups and fuzzy soft ideals, Comput. Math. Appl, 61 (2011) 255-561. https://doi.org/10.1016/j.camwa.2010.10.047 [7] F. Feng, Y. B. Jun, X. Z. Zhao, Soft semirings, Comput. Math. Appl, 56 (2005) 2621-2628. https://doi.org/10.1016/j.camwa.2008.05.011 [8] N. Kuroki, D. S. Malik, J. N. Mordeson, Fuzzy semigroups, Springer, (2003). https://doi.org/10.1007/978-3-540-37125-0 [9] D. S. Malik, J. N. Mordeson, Fuzzy Commutative Algebra, World Science publishing Co.Pte.Ltd, (1995). [10] A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl, 35 (1971) 512-517. https://doi.org/10.1016/0022- 247X(71)90199-5 [11] Maji, P.K., Biswas, R. and Roy, A.R. (2003). Soft set theory. Computers and Mathematics with Applications, 45: 555-562. [12] Molodtsov, D. (1999). Soft set Theory-First results. Computers and Mathematics with Applications, 37: 19-31. [13] Pawlak, Z. (1982). Rough sets. International Journal of Information and Computer 134 Sciences, 11: 341-356. [14] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965) 338-353. https://doi.org/10.1016/S0019- 9958(65)90241-X.