Advances in Fuzzy Mathematics. ISSN 0973-533X Volume 7, Number 1 (2012), pp. 1-9 © Research India Publications http://www.ripublication.com/afm.htm
t-Intuitionistic Fuzzy Quotient Group P.K. Sharma Post Graduate Department of Mathematics, D.A.V. College, Jalandhar City, Punjab, India E-mail:
[email protected]
Abstract In this paper, the notion of t-intuitionistic fuzzy cosets of an intuitionistic fuzzy normal subgroup and t-intuitionistic fuzzy quotient group are defined and discussed. A homomorphism from a given group onto the set of all tintuitionistic fuzzy quotient group is established. Some related results has been derived. Keywords: Intuitionistic fuzzy (IFS), Intuitionistic fuzzy subgroup (IFSG), (α, β)–Cut set, t-intuitionistic fuzzy coset, t-intuitionistic fuzzy quotient group. Mathematics Subject Classification: 03F55.
Introduction The notion of intuitionistic fuzzy set (IFS) was introduced by Atanassov [1] as a generalization of Zadeh’s fuzzy sets. IFS makes description of the objective world become more realistic, practical and accurate, making it very promising. Instead of using fuzzy approach, past researchers have studied IFSs to be applied in variety of area such as decision making[10], medical diagnostics[9] and pattern recognition[4] and seem to be more popular than fuzzy sets in recent years. Author has already introduced the notion of t-intuitionistic fuzzy cosets of an intuitionistic fuzzy module and t-intuitionistic fuzzy quotient module in [8] In this paper, we introduce the notion of t-intuitionistic fuzzy cosets of an intuitionistic fuzzy normal subgroup and tintuitionistic fuzzy quotient group and discuss some of their properties.
Preliminaries We first recall some definition for the sake of completeness of the topic under study.
2
P.K. Sharma
Definition (2.1)[1] Let X be a fixed non-empty set. An intuitionistic fuzzy set (IFS) A of X is an object of the following form A = {< x, μA (x), νA (x) >: x ∈X}, where μA: X Æ [0, 1] and νA: X Æ [0, 1] define the degree of membership and degree of nonmembership of the element x ∈X respectively and for any x ∈X, we have 0 ≤ μA (x) + νA (x) ≤ 1. Remark (2.2) (i): When μA (x) + νA (x) = 1, i.e. when νA (x) = 1-μA (x) = μAc (x). Then A is called fuzzy set. (ii) We use the notation A = (μA, νA) to denote the IFS A of X. Definition (2.3)[ 5 ]: Let G be a group. An intuitionistic fuzzy subset (IFS) A = (μA, νA) of G is called intuitionistic fuzzy subgroup (IFSG) of G if (i) μA (xy) ≥ μA (x) ∧ μA (y) (ii) μA (x-1) = μA (x) (iii) νA (xy) ≤ νA (x) ∨ νA (y) (iv) νA (x-1) = νA (x), for all x, y ∈G Or Equivalently A is IFSG of G if and only if μA (xy-1) ≥ μA (x) ∧ μA (y) and νA (xy) ≤ νA (x) ∨ νA (y) Definition (2.4)[5] An IFSG A = (μA, νA) of a group G is said to be intuitionistic fuzzy normal subgroup of G (In short IFNSG) of G if (i) μA (xy) = μA (yx) (ii) νA (xy) = νA (yx), for all x, y ∈G Or Equivalently A is an IFNSG of a group G is normal if and only if μA (y-1 x y) = μA (x) and νA (y-1 x y) = νA (x), for all x, y ∈G Definition (2.5)[5]: Let A be intuitionistic fuzzy set of a universe set X. Then (α, β)cut of A is a crisp subset Cα, β (A) of the IFS A is given by Cα, β (A) = { x: x ∈X such that μA (x) ≥ α, νA (x) ≤ β }, where α, β ∈ [0, 1] with α + β ≤ 1. Theorem (2.6) [ 5,7 ]: If A is IFS of a group G. Then A is IFSG (IFNSG) of G if and only if Cα, β (A) is a subgroup (normal) of group G, for all α, β ∈ [0,1] with α + β ≤ 1.
t-Intuitionistic fuzzy quotient group
Definition (3.1) Let A be an IFSNG of a group G. Let t ∈ [0,1]. For any x∈G Define an IFS At x of G called t – Intuitionistic fuzzy coset of A in G as follows
t-Intuitionistic Fuzzy Quotient Group
3
At x ( g ) = ( μ At ( g ) ,ν At ( g ) ), where x
for all x, g ∈ G.
x
μ A ( g ) = min{μ A ( gx ), t} and ν A ( g ) = max{ν A ( gx −1 ), 1- t} −1
t
t
x
x
Proposition (3.2) Let S be the set of all t – intuitionistic fuzzy cosets of an IFNSG A in G. i.e. S = { At x : x∈G }. Then the binary operations ⊗ defined on the set S as follows: At x ⊗ At y = At xy , for all x , y ∈ G is a well defined operation. Proof. Let At x = At x′
At y = At y′ , for some x , y , x ′ , y ′ ∈ G
and
Let g∈G be any element, then [ A t x ⊗ A t y ]( g ) = ( A t x y )( g ) = ( μ A t ( g ) ,ν xy
A t xy
( g ))
Now μ At ( g ) = min{μ A g ( xy )−1 , t} = min{μ A ( gy −1 ) x−1 , t} = μ At ( gy −1 ) = μ At ( gy −1 ) xy
x′
x
= min{μ A ( gy ) x′ , t} = min{μ A ( x′ g ) y , t} = μ At ( x′ g ) = μ At ( x′−1 g ) −1
−1
−1
−1
−1
y′
y
= min{μ A ( x′ g ) y′ , t} = min{μ A y′ ( x′ g ), t} = min{μ A ( y′ x′ ) g , t} −1
−1
−1
−1
−1
−1
= min{μ A ( x′y′)−1 g , t} = min{μ A g ( x′y′)−1 , t} = μ At ( g ) x′y′
Similarly, we can show that ν At ( g ) = ν At ( g ) , ∀ g ∈ G. xy
x′y′
Therefore ⊗ is well defined operation on S. Lemma (3.3): If A is IFNSG of a group G. Then At x = At x′ ⇔ Nx = Nx ′ , for x , x ′ ∈ G , wherer N = Ct ,1−t ( A) Proof. Let At x = At x ′
( min{μ
A
for x , x ′ ∈ G , then
At x ( x ′) = At x ′ (x ′)
( x ′x −1 ), t}, max{ν A (x ′x −1 ) ,1 - t}) = ( min{μ A ( x ′x ′−1 ), t}, max{ν A (x ′x ′−1 ) ,1 - t}) = ( min{μ A (e), t}, max{ν A (e) ,1 - t})
⇒ min{μ A ( x ′x ), t} = t
= (t ,1- t) and max{ν A (x ′x −1 ) ,1 - t} = 1 − t
⇒ μ A ( x ′x −1 ) ≥ t
ν A (x ′x −1 ) ≤ 1 - t and so x ′x −1 ∈ Ct ,1− t ( A) = N
−1
⇒ N x = N x′
and
................................................................................................................(*)
Now, we show that if N x = N x′, then At x = At x′ A x (y) ≠ A x′ (y) t
Let for some y∈G,
t
i.e. ( min{μ A ( xy −1 ), t}, max{ν A (xy −1 ) ,1 - t}) ≠ ( min{μ A ( x′y −1 ), t}, max{ν A (x ′y −1 ) ,1 - t})
4
P.K. Sharma Suppose μA (x y-1) < t and μA (x′ y-1) ≥ t Therefore νA (x y-1) ≤ 1-t and νA (x′ y-1) ≤ 1-t ⇒ x′ y-1 ∈Ct, 1-t (A) = N ⇒ N x′ y-1 = N ⇒ Nx y-1 = N (Using (*)) ⇒ x y-1 ∈ N and so μA (x y-1) ≥ t, a contradiction
Similarly, if μA (x y-1) ≥ t and μA (x′ y-1) < t also leads to contradiction Therefore either μA (x y-1) ≥ t and μA (x′ y-1) ≥ t i.e. νA (x y-1) ≤ 1-t and νA (x′ y-1) ≤ 1-t or μA (x y-1) < t and μA (x′ y-1) < t i.e. νA (x y-1) ≤ 1-t and νA (x′ y-1) ≤ 1-t In the first case Min { μA (x y-1), t } = t and max { νA (x y-1), 1-t } = 1-t And so A t x ( y ) = (t , 1 − t ) and also Min { μA (x′ y-1), t }= t and max { νA (x′ y-1), 1-t } = 1-t And so A t x′ ( y ) = (t , 1 − t ) . Thus A t x ( y ) = A t x′ ( y ) , for all y ∈ G Therefore A t x = A t x′ In the second case Min { μA (x y-1), t } = μA (x y-1) < t and max { νA (x y-1), 1-t } = 1-t And also Min { μA (x′y-1), t }= μA (x′ y) < t and max { νA (x′ y-1), 1-t } = 1-t Now since N x = N x′, therefore let x = n x′, where n ∈ N So that μA (n) ≥ t and νA (n) ≤ 1 – t A t x′ ( y ) = ( min{μ A ( yx′−1 ), t}, max{ν A (yx′−1 ) ,1 - t}) = ( μ A ( yx −1n), 1- t ) = ( μ A (nyx −1 ), 1- t ) ≥ ( μ A (n) ∧ μ A ( yx −1 ), 1- t ) = ( μ A ( yx −1 ), 1- t ) =
( min{μ
A
( yx −1 ), t}, max{ν A (yx −1 ) ,1 - t})
= At x ( y ) Thus A t x′ ( y ) ≥ A t x ( y )
, for all y ∈ G
t-Intuitionistic Fuzzy Quotient Group Similarly
5
A t x ( y ) = ( min{μ A ( yx −1 ), t}, max{ν A (yx −1 ) ,1 - t}) = ( μ A (yx −1 ), 1- t ) = ( μ A ( yx′−1n −1 ), 1- t )
≥ ( μ A (n) ∧ μ A ( yx′−1 ), 1- t ) = ( μ A ( yx′−1 ), 1- t ) =
( min{μ
A
( yx′−1 ), t}, max{ν A (yx′−1 ) ,1 - t})
= A t x′ ( y ) Thus A t x ( y ) ≥ At x′ ( y )
, for all y ∈ G
Therefore At x = A t x′ Hence the result proved. Proposition (3.4) The set S of all t-Intuitionistic fuzzy cosets of an IFNSG A of a group G, form a group under the well-defined operations ⊗. Proof. It is easy to check that the identity element of S is At e , where e is the identity element of group G, and the inverse of an element At x is At x−1 . Proposition (3.5) The IFS B of S defined by B(A t a ) = ( μ B (A t a ) , ν B (A t a ) ) Where μ B (A t a ) =
Sup {μ A ( x ) : x ∈ G} and ν B (A t a ) =
At x = At a
Inf {ν A ( x) : x ∈ G} is
At x = A t a
a IFSG of S, called t-Intuitionistic fuzzy quotient group. Proof. Let a, b ∈G and let B(At a ) = ( θ1 , θ 2 ) and B(At b ) = ( φ1 , φ2 ) , where
θ1 = Sup {μ A ( x) : x ∈ G } , θ 2 = Inf {ν A ( x) : x ∈ G } and A t x = At a
At x = At a
φ1 = Sup {μ A ( x) : x ∈ G } , φ2 = Inf {ν A ( x) : x ∈ G } A t x = At b
A t x = At b
∃ x, y ∈ G such that θ1-∈ < μA (x), Nx = Na and φ1-∈ < μA (y), Ny = Nb Therefore Nxy = Nab ⇒ At xy = At ab ⇒ At x ⊗ At y = A t a ⊗ A t b So that μ A ( xy ) ≤ μ B (At x ⊗ At y ) = μ B (At a ⊗ At b ) Now μ A ( xy ) ≥ μ A ( x) ∧ μ A ( y ) = μ A ( x)[ say ] > θ1 - ∈
∴ θ1 - ∈ < μ A ( xy ) ≤ μ B (At a ⊗ At b ) , ∀ ∈ > 0 so that θ1 ≤ μ B (At a ⊗ At b )
6
P.K. Sharma Now two cases arises Case(i) When θ1 ≥ φ1
, then θ1 − ∈ ≥ φ1 - ∈
φ1 - ∈ ≤ θ1 − ∈ ≤ μ B (At a ⊗ A t b ) , ∀ ∈ > 0 so that φ1 ≤ μ B (At a ⊗ A t b ) Therefore μ B (A t a ) ∧ μ B (A t b ) = θ1 ∧ φ1 = φ1 ≤ μ B (A t a ⊗ At b ) Case(ii) When θ1 < φ1 , then
μ B (At a ) ∧ μ B (At b ) = θ1 ∧ φ1 = θ1 ≤ μ B (At a ⊗ At b ) Thu s in any case we get μ B (At a ⊗ A t b ) ≥ μ B (At a ) ∧ μ B (At b ) Similarly, we can show that ν B (A t a ⊗ A t b ) ≤ ν B (A t a ) ∨ ν B (A t b ) Next to show that B(At a ) = B((At a )−1 ) = B(At a−1 ) Since A t x = A t a ⇒ Nx = Na and At y = At a−1 ⇒ Ny = Na-1 = (Na)-1= (Nx)-1=Nx-1 yx ∈ N ⇒ Nyx = N ⇒ At yx = At e ⇒ At y ⊗ At x = At e so At y = At x−1 and we know that μA (x-1) = μA (x). Thus μ B (At a ) = μ B (At a−1 ) . Similarly, we van show that ν B (At a ) = ν B (At a−1 ) . Whence B(At a ) = B(At a−1 ) . Proposition (3.6) A mapping f: G Æ S, where G is a group and S is the set of all tintuitionistic fuzzy cosets of the IFNSG A of G defined by f (x) = At x , is an onto homomorphism with ker f = N (= Ct,1-t (A)), where t ∈[0,1]) Proof. Clearly f is an onto homomorphism Let x ∈ ker f, then f (x) = identity element of S = At e Therefore At x = At e ⇒ ker f ⊆ N
so
⇒
Nx=Ne=N
x∈ N
(1)
Conversely, let x ∈ N ⇒ Nx = N so that N xg-1 = Ng-1 ∀ g ∈ G. If possible let x ∉ kerf i.e. At x ≠ At e
therefore there
exists g ∈ G such that At x ( g ) ≠ At e (g) Suppose μ A ( xg −1 ) < t Therefore μ A ( g i.e.
Nxg
−1
=N
−1
and μ A ( g −1 ) ≥ t , i.e. ν A ( xg −1 ) ≤ 1- t
) ≥ t and ν A ( g ⇒
xg
−1
−1
⇒
∈ N and so μ A ( xg
Similarly , μ A ( xg ) ≥ t and μ A ( g −1
∴ either μ A ( xg ) ≥ t , μ A ( g −1
) ≤ 1- t
−1
or μ A ( xg −1 ) < t , μ A ( g −1 ) < t
−1
−1
g
−1
and ν A ( g
∈ N so Ng
−1
−1
) ≤ 1- t
=N
) ≥ t , a contradiction
) < t is not possible .
) ≥ t i.e. ν A ( xg −1 ) ≤ 1- t and ν A ( g −1 ) ≤ 1- t i.e. ν A ( xg −1 ) ≤ 1- t and ν A ( g −1 ) ≤ 1- t
t-Intuitionistic Fuzzy Quotient Group
7
In the first case Min { μA (xg-1), t } = t and Max { νA (xg-1), 1-t } = 1-t and so At x ( g ) = (t ,1 − t )
similarly we get At e ( g ) = (t ,1 − t ) . Thus At x ( g ) = At e ( g ) In the second case Min { μA (x g-1), t } = μA (xg-1) < t and Max { νA (xg-1), 1-t } = 1-t At x ( g ) = ( min{ μ A ( x g −1 ) , t} , max { ν A ( x g −1 ) , 1- t }) = ( μ A ( x g −1 ) , 1-t ) ≥ ( μ A ( x ) ∧ μ A (g) , 1- t) = ( μ A ( g) , 1 -t )
[ ∵ x ∈ N ∴ μ A ( x ) ≥ t an d μ A ( g) = μ A ( g
= ( min{ μ A (eg ) , t} , max { ν A ( eg −1
−1
−1
) < t]
) , 1- t })
t
= A e (g) = ( μ A ( g −1 ) , 1 -t ) = ( μ A ( xg −1 x −1 ) , 1 -t )
[ As A is IFNSG of G so μ A ( xg −1 x −1 ) = μ A ( g −1 ) ]
≥ ( μ A ( x g −1 ) ∧ μ A ( x ) , 1- t ) = ( μ A ( x g −1 ),1 − t ) = ( min{ μ A ( x g −1 ) , t} , max { ν A ( x g −1 ) , 1- t }) = At x ( g )
f ( x) = Identity element of S
i.e. ∴
N ⊆ ker f
so
and so x ∈ ker f
ker f = N.
Proposition (3.7) If f: G Æ S, is an onto homomorphism, then f (A) = B, where A is IFS of G and B is IFS of S. Proof. Let At a ∈ S be any element of S , where a ∈ G such that f (a ) = At a Let A be IFS of G , then ⎧⎛ ⎞ −1 t −1 t ⎪⎜ Sup{μ A ( x) : x ∈ f ( A a )}, Inf{ν A ( x) : x ∈ f ( A a )} ⎟ f ( A)( A a ) = ⎨⎝ ⎠ ⎪ ( μ A (e) , ν A (e) ) ⎩ t
⎧⎛ ⎞ t t t t ⎪⎜ Sup{μ A ( x) : A x = A a }, Inf{ν A ( x) : A x = A a } ⎟ = ⎨⎝ ⎠ ⎪ ( μ A ( e) , ν A ( e) ) ⎩ t = B(A a ) Hence f (A) = B Theorem (3.8): Let A be a IFNSG of G and B be a IFSG of S, then Ct ,1−t ( B) = { At e }
8
P.K. Sharma
Proof. Now B( At e ) = ( μ B ( At e ) , ν B ( At e ) ) , where
μ B ( At e ) = Sup{μ A ( x) : N x = N } = Sup{μ A ( x) : x ∈ N} ≥ μ A (n) , for all n ∈ N = Ct ,1−t ( A) ≥ t Similarly , we can show that ν B ( At e ) ≤ 1 -t . Thus At e ∈ Ct ,1−t ( B)
⇒
μ B ( At a ) ≥ t
and ν B ( At a ) ≤ 1-t
Let
At a ∈ Ct ,1−t ( B)
Let
θ1 = μ B ( At a ) = Sup{μ A ( x) : N x = Na } and θ 2 = ν B ( At a ) = Inf {ν A ( x) : N x = Na}
Therefore θ1 ≥ t and θ2 ≤ 1-t. Let ε > 0 be given, ∃’s x, y ∈ G such that N x = N a so that x a-1 = n1 ∈ N and μA (x) > θ1-ε ≥ t-ε and N y = N a so that y a-1 = n2 ∈ N and νA (y) < θ2 + ε ≤ (1-t) + ε if μ A (an1 ) ≥ t ⎧ ≥t μ A (a) = μ A (an1n−11 ) ≥ μ A (an1 ) ∧ μ A (n1 ) = ⎨ and ⎩= μ A (an1 ) if μ A (an1 ) < t if ν A (an2 ) ≤ 1 − t ⎧ ≤ 1− t ν A (a ) = ν A (an2 n −12 ) ≤ ν A (an2 ) ∨ ν A (n2 ) = ⎨ ⎩= ν A (an2 ) if ν A (an2 ) > 1 − t Thus in any case μA (a) > t-ε and νA (a) < (1-t) + ε, for all ε > 0 ⇒ μA (a) ≥ t and νA (a) ≤ (1-t) implies that a ∈ Ct, 1-t (A) ⇒ N a = N so At a = At e Hence Ct ,1−t ( B ) = { At e } .
References [1] K.T. Atanassov, “Intutionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, 1986, pp. 87–96 [2] K. T. Atanassov, “New operations defined over the intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 61, no. 2, 1994, pp. 137–142 [3] L. Atanassov, “On Intuitionistic fuzzy versions of L. Zadeh’s extension principle “, Notes on intuitionistic Fuzzy Sets, (13) (3), 2006, 33-36 [4] Dengfeng Li, Cheng Chutian, “ New similarity measure of IFS and application to pattern recognition”, J. pattern recognition letter, Vol. 23, 2002, 221-225 [5] Sharma, P.K., “ (α, β)-Cut of Intuitionistic fuzzy groups” International Mathematics Forum, Vol. 6, 2011, no. 53, 2605-2614 [6] Sharma, P.K., “Homomorphism of Intuitionistic fuzzy groups ” International Mathematics Forum, Vol. 6, 2011, no. 64, 3169-3178
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[7] Sharma, P.K., “ On the direct product of Intuitionistic fuzzy groups”, International Mathematical Forum, Vol. 7, 2012, no. 11, 523-530 [8] Sharma, P.K., “ t-Intuitionistic fuzzy Quotient modules” International Journal of Fuzzy Mathematics & System (IJFMS), Vol., 2, No. 1, 2012, 37-42 [9] Supriya K.De, R.Biswas A. R. Roy, “ An application of Intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and System, 117 (2001), 209-213 [10] Szmidt E and Kacprzyk J. “ Intuitionistic fuzzy set in group decision making, Notes IFS 2 (1), 1996, 11-14 [11] Zadeh,L.A., “ Fuzzy Sets”, Information and Control, 8, (1965), 338-353
