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The lower and upper approximations in a fuzzy group
Intelligent Systems
NORTH-HOLLAND
The Lower and Upper
Approximations
in a Fuzzy Group
NOBUAKI KUROKI Department of Mathematics, Joetsu University of Education, Joetsu-shi, Niigata-kenn, Japan 943 and PAUL P. WANG Department of Electrical Engineering, Duke University, Box 90291, Durham, North Carolina 27708-0291
ABSTRACT' In this paper, we shall introduce the notion of a rough subgroup with respect to a normal subgroup of a group, and give some properties of the lower and the upper approximations in a group. Also, we will discuss a rough subgroup with respect to a t-level subset of a fuzzy normal subgroup.
1.
INTRODUCTION
T h e notion of rough sets was introduced by Pawlak in his paper [5]. Some authors, for example, Bonikowaski [2], Iwinski [3], and P o m y k a l a and P o m y k a l a [7], studied algebraic properties of rough sets. But there was no paper on rough sets in algebraic structures. Recently, Biswas and N a n d a [1] gave the notion of rough subgroups. Because their notion depends on the u p p e r approximation and does not depend on the lower approximation, we will discuss the lower and the upper approximation of a group and of a fuzzy group in detail. 2.
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Let G be a group (as universe) with identity e. Let ~ be a congruence relation of G, t h a t is, $ is an equivalence relation on G such t h a t (a,b) e 6 ( a , b e G )
implies
(ax, bx) e 6
and
(xa, xb) e
for a l l x o f G . A subgroup N of G i s called normal if a N = N a for all a of G. Then, as is well known and easily seen, there exists a one-to-one
INFORMATION SCIENCES 90, 203-220 (1996) (~ Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0020-0255/96/$15.00 SSDI 0020-0255(95)00282-0
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correspondence between the set of all congruence relations on G and the set of all normal subgroups of G. Therefore, we can identify the notion of normal subgroups and of congruences on a group. Let N be a normal subgroup of G. Let A be a nonempty subset of G. Then the sets
N_(A) = { x • G : x N C A }
and
N^(A) = { x c G : x N n A ~ O }
are called, respectively, lower and upper approximations of a set A with respect to the normal subgroup N. Throughout this paper, H and N mean normal subgroups of G. The following is due to Proposition 2.2 of [5]. We shall give a proof for completeness. PROPOSITION 2.1. Let N and H be normal subgroups of a group G. Let A and B be any nonempty subsets of G. Then (1) (2) (3) (4) (5) (6) (7) (8)
N_(A) C A C N^(A). N^(A U B) = N^(A) U N^(B). N_ ( A n B) -- N_ (A) n N_ (B). A C_ B implies N_ (A) c_ N_ (B). A C_ B implies N^(A) C_N^(B). N_ (A u B) _~ N_ (A) U N_ (B). N^(A N B) C_ N^(A) n N^(B). N c_ H implies N"(A) C_H^(A).
Proof. (1) I f V a c N_(A), then a = ae e aN C_ A. T h u s N _ ( A ) C A . Next, ifVa E A, then a = ae C aN. Thus a E a N N A , that is, a N N A ~ O. This implies a E N^(A), and so A C N^(A). (2) aEN^(AUB)
¢:> a N N ( A U B ) ~ O tee ( a N N A ) U ( a N N B ) ~O ¢ez a N N A ~ O or a N N B ~ O ¢:> a E N^(A) or a c N^(B) a • N^(A) u N^(B).
Thus N^(A U B) = N^(A) U N^(B).
(3) a • N _ ( A N B ) ~ aN aN ¢=> a • ¢:> a E
C ANB C A and aN C_ B N_(A) and a • N _ ( B ) N_ (A) N N_ (B).
Thus N^(A n B) = N_ (A) N N_ (B).
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(4) Since A C_ B, A n B = A. Then by (3), we have N_ (A) = N= ( A n B) = N_ (A) N N_ (B), which yields N_ (A) C_ N_ (B). (5) Since A _c B, A U B = B. Then by (2), we have
N^(A) = N^(A U B) = N^(A) U N^(B), which yields N^(A) C_ N^(B). (6) A C A U B a n d B C _ A u B ; b y ( 4 ) , w e h a v e N_(A) C _ N _ ( A u B )
and
N_(B) CN_(AUB).
Thus N_(A) U N_(B) C_ N _ ( A U B). (7) A o B C_ A and A n B C_ B; by (g), we have
N^(A N 13) C_N^(A)
and
N^(A O B) C_N^(B).
Thus N^(A N B) C N"(A) N N^(B). (8) If Vc E N^(A), then 3x E cN n A, and so x E cN C_ cH. Therefore, x E cH n A. This implies c E H^(A), and N^(A) C H^(A). This completes the proof. [] PROPOSITION 2.2. Let N be a normal subgroup of a group G. Let A and B be nonempty subsets of G. Then
N^(A)N^(B) = N^(AB). Proof. Let c be any element of N^(AB). Then cN N A B ¢ O. Thus there exists an element x in G such that z E cN N A B , and so x E cN and zEAB. Thenz=abwithaEAandbE B. S i n c e c E x N = (ab),N= (aN)(bN), we have c = yz with y E aN and z E bN. Then a E yN, and so a E y N n A. Thus y E N^(A). Similarly we have z E N^(B). Thus c = yz E N^(A)N^(B), and so we have N^(AB) C_ N^(A)N^(B). Conversely, let c be any element of N^(A)N^(B), then c = ab with a E N^(A) and b E N^(B). Thus there exist elements x and y in G such that x E aN n A and y E bN N B, and so x E aN, x E A, y E bN, and y E B. Since N is normal, xy E (aN)(bN) = abN, and xy E AB. Thus xy E abN n AB, which yields that c = ab E N^(AB), and so N ^ ( A ) N ^ ( B ) C_ N"(AB). Therefore we have N^(A)N^(B) = N^(AB). This completes the proof. [] PROPOSITION 2.3. Let N be a normal subgroup of a group G. Let A and B be nonempty subsets of G. Then
N_ (A)N_ (B) g N_ (AB).
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Proof. Let c be any element of N _ ( A ) N _ ( B ) . Then c = ab with a E N _ (A ) and b E N_(B). Thus aN C A a n d b N C_ B. S i n c e N i s n o r m a l , cN = abN = (aN)(bN) C_ AB, and so c E N _ ( A B ) . Thus N _ ( A ) N _ ( B ) C_ N_ (AB), which completes the proof. [] REMARK 1. Let H and N be normal subgroups of a group G. Then, as is well known and easily seen, H n N is also a normal subgroup of G. PROPOSITION 2.4. Let H and N be normal subgroups of a group G. Let A be a nonempty subset of G. Then
(H N N)^(A) = H^(A) n N^(A). Proof. VcE ( H N N ) ^ ( A ) ~ c ( H N N ) N A #O ¢* 3 a E c ( H N N ) N A aEc(HNN)
and
aEA
and
aEcN, aEA
¢=~ a E cH, a E A 3aEcHNA ~=~ c E H^(A)
and and
3aEcNNA cEN^(A)
c E H^(H) n N^(A). Thus we obtain that (H n N)^(A) = H^(A) n N^(A). This completes the proof. PROPOSITION 2.5. Let H and N be normal subgroups of a group G. Let A be a nonempty subset of G. Then (H n N ) _ (A) = H_ (A) n N_ (A).
Proo/. VcE ( H N N ) _ ( A ) ~=~ c ( H N N ) C A ~ cH C_ A ¢~ c E H _ ( A )
and
cN C_ A
and
cEN_(A)
¢~ c E H_ (A) n N_ (d). Thus we have ( H N N ) _ ( A ) -- H _ ( A ) n N_(A). proof.
This completes the []
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N ( A ) = ( N _ ( A ) , N ^ ( A ) ) is called a r o u g h set of A in G. A n o n e m p t y s u b s e t A of a g r o u p G is called an N ^ - r o u g h [normal] s u b g r o u p of G if t h e u p p e r a p p r o x i m a t i o n N ^ ( A ) of A is a [normal] s u b g r o u p of G. Similarly, a n o n e m p t y s u b s e t A of G is called an N _ - r o u g h [normal] s u b g r o u p of G if N _ (A) is a [normal] s u b g r o u p of G. PROPOSITION 3.1. Let N be a normal subgroup of a group G. If A is a subgroup of G, then it is an N^-rough subgroup of G.
Proof. Let e be t h e i d e n t i t y of G. Since N a n d A are s u b g r o u p s of G , e E A a n d e = ee E eN, a n d s o e E e N A A . T h u s e N N A ~ 9 . T h i s implies t h a t e E N ^ ( A ) . L e t a a n d b be any e l e m e n t s of N^(A). T h e n t h e r e exist e l e m e n t s x a n d y in G such t h a t x E a N N A a n d y E b N N A . T h u s x E aN, y E bN a n d x E A, y E A. Since A is a s u b g r o u p of G, xy E A. A n d since N is a n o r m a l s u b g r o u p of G, xy E (aN)(bN) = abN. T h u s xy E abN N A, a n d so ab E N " ( A ) . L e t a be a n y e l e m e n t of N^(A). T h e n x E a N A A for some x E G, t h a t is, x E a N a n d x E A. T h e n s i n c e A is a s u b g r o u p of G, x -1 E A. O n t h e o t h e r h a n d , since x = ah for some h E N , a n d since N is a n o r m a l s u b g r o u p of G, h -1 E N and we have x -1 = (ah) -1 = h - l a -1 E N a -1 = a - i N . T h u s x -1 E a - I N n A , a n d so a -1 E N^(A). T h e s e i m p l y t h a t N " ( A ) is a s u b g r o u p of G. This c o m p l e t e s t h e proof. [] PROPOSITION 3.2. Let N be a no~nal subgroup of a group G. If A is a normal subgroup of G, then it is an N^-rough normal subgroup of G.
Proof. It suffices to show t h a t N " ( A ) is normal. Let a a n d x be a n y e l e m e n t s of N ^ ( A ) a n d G, respectively. T h e n t h e r e exists an e l e m e n t y in G such t h a t y E a N A A, t h a t is, y E a N a n d y E A. Since N is n o r m a l , xyx -1 E
x ( a N ) x -1 = ( x a ) ( N x -1) = ( x a ) ( x - l N ) : ( x a x - 1 ) N .
Since A is n o r m a l ,
x y x - 1 E x A x - 1 C A. T h u s x y x -1 E ( x a x - 1 ) N N A, and so xax -1 E N^(A). N ^ ( A ) is n o r m a l , which c o m p l e t e s t h e proof.
This means that []
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PROPOSITION 3.3. Let N be a normal subgroup of a group G. If A is a subgroup of G such that N C A, then it is an N_-rough subgroup of G.
Proof. Since e N = N C A, e • N_ (A). Let a and b be any elements of N_ (A). Then a N C_ A and bN C_ A. Since N is a normal subgroup and A is a subgroup of G, abN = (aN)(bN) C_ A A C_ A. This implies that ab • N_ (A). Let a be any element of N_ (A). Then
a=ae•aNCA. Since A is a subgroup of G, a -1 • A. Thus we have
a - i N C A A C_ A. This implies that a -1 • N _ ( A ) . which completes the proof.
Therefore, N _ ( A ) is a subgroup of G, []
PROPOSITION 3.4. Let N be a normal subgroup of a group G. If A is a normal subgroup of G such that N C_ A, then it is an N_-rough normal subgroup of G.
Proof. It suffices to show that N_ (A) is normal. Let a and x be any element of N_ (A) and G, respectively. Then, a N C_ A. Since N and A are normal, ( x a x ) - l g = x ( a N ) x -1 C_ x A x -1 C_ A, and so xax -1 • N _ ( A ) , which means that N _ ( A ) is normal. This completes the proof. [] REMARK 2. The product H N of normal subgroups H and N of a group G is also a normal subgroup of G. So we can consider the lower and upper approximations ( H N ) _ (A) and ( H N ) ^ ( A ) for a nonempty subset A of G. PROPOSITION 3.5. Let H and N be normal subgroups of a group G. If A is a subgroup of G, then
H^(A)N^(A) C (HN)^(A). Proof. Let c be any element of H ^ ( A ) N ^ ( A ) . Then c = ab with a • HA(A) and b • N " ( A ) . Then there exist elements x and y in G such that x • a H A A and y • b N A A . T h u s x • aH, y • bN, x • A, and y • A. Then, since H is normal,
x y • (aH)(bN) = {a( H b ) N } = { a ( b H ) N } = { (ab)H } N = (ab)H N = cH N.
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Since A is a s u b g r o u p of G, xy E A. Therefore, xy E c H N N A, a n d so we have c E ( H N ) ^ ( A ) . T h u s H ^ ( A ) N ^ ( A ) C_ ( H N ) ^ ( A ) . T h i s c o m p l e t e s t h e proof. [] PROPOSITION 3.6. Let H and N be normal subgroups of a group G. If A is a subgroup of G, then
( H N ) ^ ( A ) C H ^ ( A ) N (1 N ^ ( A ) H . Proof. Let c be a n y element of ( H N ) ^ ( A ) . T h e n t h e r e exists an e l e m e n t x in G such t h a t x c c ( H N ) A A. T h u s x E c H N a n d x C A. T h u s x = cab w i t h a E H a n d b E N . Note t h a t a -1 E H a n d b -1 C N. Since H is n o r m a l , x = cab E crib = cbH, a n d so x E cbH N A. T h u s cb E H ^ ( A ) , a n d so c c H^(A)b -1 c_ H ^ ( A ) N . Similarly, c E N ^ ( A ) H . Therefore, c e H ^ ( A ) N A N ^ ( A ) H , and so ( H N ) ^ ( A ) C_ H ^ ( A ) N A N ^ ( A ) H , which c o m p l e t e s t h e proof. [] PROPOSITION 3.7. Let H and N be normal subgroups of a group G. If A is a subgroup of G, then
H _ ( A ) N _ ( A ) c_ ( H N ) _ ( A ) . Proof. Let c be a n y element of H _ ( A ) N _ ( A ) . T h e n c = ab w i t h a E H _ (A) a n d b E N _ (A). T h u s a H C_ A a n d bN C_ A. Since H is a n o r m a l s u b g r o u p a n d A is a s u b g r o u p of G, we have ( a b ) H N = { a ( b H ) } N = { a ( H b ) } N = { ( a H ) b } N = (aH)(bN) C_ A A C_ A. Therefore, we have c = ab e ( H N ) _ (A), a n d so H_ (A)N_ (A) C_( H N ) _ (A), which c o m p l e t e s t h e proof. []
4.
R O U G H S U B G R O U P S IN A F A C T O R G R O U P
T h e lower a n d u p p e r a p p r o x i m a t i o n s can be p r e s e n t e d in a n equivalent form as shown below. Let N be a n o r m a l s u b g r o u p of a g r o u p G, a n d A a n o n e m p t y s u b s e t of G. T h e n
N_(A) = {aN e G/N:
a N C A},
N^(A)={aNeG/N:
aNnA~O}.
PROPOSITION 4.1. Let N be a normal subgroup of a group G. If A is a subgroup of G, then N ^ ( A ) is a subgroup of G / N .
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Proof. N o t e t h a t N = e N is t h e i d e n t i t y of G / N . Since e E N = e N a n d e E A , e E e N C ~ A . T h u s N = e N E N ^ ( A ) . Let a N a n d bN b e a n y e l e m e n t s of N ^ ( A ) . T h e n t h e r e exist e l e m e n t s x a n d y in G such t h a t x E aNAA and y E b N n A . T h u s x E a N , y E bN, x E A, a n d y E A. T h e n , since N is normal, x y E ( a N ) ( b N ) = (ab)N. Since A is a s u b g r o u p of G, x y E A. T h u s x y E (ab)N A A. T h e r e f o r e we have ( a N ) ( b N ) = (ab)N E N ^ ( A ) . Let a N be a n y e l e m e n t of N ^ ( A ) . T h e n t h e r e exists an e l e m e n t x in G such t h a t x E a N A A. T h u s x E a N a n d x E A. T h u s x = an for some n E N . Note t h a t ( a N ) -1 = a - I N . T h e n we have x -1 = (an) -1 = n - l a -1 E N a -1 = a - i N . Since x -1 E A, x -1 E a - i N N A, a n d so a - i N E N ^ ( A ) . Therefore, we o b t a i n t h a t N ^ ( A ) is a s u b g r o u p of G / N , which c o m p l e t e s t h e proof. [] PROPOSITION 4.2. Let N be a normal subgroup of a group G. If A is a normal subgroup of G, then N ^ ( A ) is a normal subgroup of G / N .
Proof. Let a N a n d x N be a n y e l e m e n t s of N ^ ( A ) a n d G / N , respectively. T h e n t h e r e exists an e l e m e n t y in G such t h a t y E a N A A. T h u s y E a N a n d y E A. Since N is normal, we have x y x -1 E x ( a N ) x -1 = ( x a x - 1 ) N . A n d since A is also normal,
x y x - 1 E x A x - 1 C_ A. T h u s x y x -1 E ( x a x - 1 ) N N A, and so
( x N ) ( a N ) ( x N ) -1 = ( x N ) ( a N ) ( x - l g ) = ( x a x - 1 ) N E N ^ ( A ) . T h e n it follows from this and P r o p o s i t i o n 4.1 t h a t N ^ ( A ) is a n o r m a l s u b g r o u p of G I N . T h i s c o m p l e t e s t h e proof. [] PROPOSITION 4.3. Let N be a normal subgroup of a group G. I f A is a normal subgroup of G such that N C A, then N_ (A) is a subgroup of G/N.
Proof. Since e N = N C A, e N E N_ (A). Let a N a n d bN b e a n y e l e m e n t s of N _ ( A ) ; t h e n a N C A and bN C A. Since N is n o r m a l , abN = ( a N ) ( b N ) C A A C_ A,
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and so we have
( a N ) ( b N ) = ab(N) C N _ ( A ) . Let a N be any element of N_(A). Then, since a = ae • a N C A, a -1 E A. Thus
a - i N C_ A A C_ A. This means that
(aN) -1 = a - i N C N _ ( A ) . Thus N_ (A) is a subgroup of G / N , which completes the proof.
[]
PROPOSITION 4.4." Let N be a normal subgroup of a group G. If A is a normal subgroup of G such that N C_ A, then N_ (A) is a normal subgroup
ola/N. Proof. Let a N and x N be any elements of N _ ( A ) and G / N , respectively. Then a N C_ A. Since N and A are normal, ( x a x - 1 ) N = ( x a ) ( x - l N ) = ( x a ) ( N x -~) = x ( a N ) x -~ C_ x A x -1 C A. This implies that
( x N ) ( a N ) ( x N ) -1 = ( x N ) ( a N ) ( x - l N )
= (xax-1)N E N_(A).
Then it follows from this and Proposition 4.3 that N _ ( A ) is a normal subgroup of G / N . This completes the proof. []
5.
H O M O M O R P H I C IMAGES OF THE U P P E R A P P R O X I M A T I O N
Let G and G ~ be two groups. A mapping f: G --~ G ~ is called a homomorphism from G to G t if
f(ab) = f ( a ) f ( b ) for all a, b E G. We denote by e ~ the identity of G( Then the set Ker(f) = {x • G: f ( x ) = e'} is called the Kernel of f. group of G.
As is well known, Ker(f) is a normal sub-
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LEMMA 5.1. Let f be a homornorphism of a gwup G to a group G'. If A is a nonempty subset of G, then
f(A) ~ f(A Ker(f)). Proof. L e t y be any element of f ( A ) . T h e n f(a) = y for s o m e a E A. W e n o t e t h a t e E K e r ( f ) . T h u s we have y = f(a) = f(ae) E f ( A K e r ( f ) ) , a n d so f ( A ) c_ f ( A K e r ( f ) ) . Conversely, let y be a n y e l e m e n t of f ( A K e r ( f ) ) . T h e n f(a) = y for some a E A K e r ( f ) . T h u s a = b c w i t h b E A a n d c ¢ Ker(f). Then
y = f(a) = f(bc) = f(b)f(c) = f(b)e I = f(b) E f ( A ) , a n d so f ( A K e r ( f ) ) C_ f ( A ) . Therefore, f ( A ) = f ( A K e r ( f ) ) . PROPOSITION 5.2. Let f be a homomorphism of G to G r, and N a normal subgroup of G. If A is a nonempty subset of G, then
f ( A ) c_ f ( N ^ ( A ) ) c_ f ( A N ) . Proof. B y P r o p o s i t i o n 2.1(1), A C_ N^(A), a n d since f is a h o m o m o r p h i s m of G t o G ' , we have f ( A ) C f ( N ^ ( A ) ) . To see I ( N ^ ( A ) ) C f ( A N ) , let y be a n y e l e m e n t of f ( N ^ ( A ) ) ; t h e n f(a) = y for some a E N^(A). T h u s t h e r e exists an e l e m e n t x in G such t h a t x E aN N A. T h e n x E a N a n d x E A. T h u s x = ab for some b E N , t h a t is, a = xb -1. Since N is a s u b g r o u p of G, b - i E N . T h e n we have y = f(a) = f(xb -1) E f ( A N ) , []
a n d so f ( N ^ ( A ) ) c_ f ( A N ) , which c o m p l e t e s t h e proof.
PROPOSITION 5.3. Let f be a homomorphism of G to G I. If A is a nonernpty subset of G, then
f(A) = f(Ker(f)^(A)). Pro@ Since K e r ( f ) is a n o r m a l s u b g r o u p of G, L e m m a 5.1 a n d P r o p o s i t i o n 5.2 t h a t
it follows from
f ( A ) c_ f ( K e r ( f ) ^ ( A ) ) c_ f ( A K e r ( f ) ) = f ( A ) . Therefore, we o b t a i n t h a t f ( A ) = f ( K e r ( f ) ^ ( A ) ) , which c o m p l e t e s t h e proof.
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6. T H E L O W E R A N D U P P E R A P P R O X I M A T I O N S ON A F U Z Z Y NORMAL SUBGROUP A fuzzy subset # of a group G is called a fuzzy subgroup of G if the following conditions hold: (1) #(ab) > min{#(a),#(b)} for all a,b ~ G, (2) , ( a -1) _> s(a) for all a E G, (3) s ( e ) = 1. A fuzzy s u b g r o u p S of G is called a fuzzy subgroup of G if (4) s(ab) = s(ba) for all a. b E G. Let S be a fuzzy normal subgroup of G. For each t E [0, 1], the set St = { ( a , b ) E G x G : s ( a b
-1) >_t}
is called a t-level relation of S. T h e n we have the following. LEMMA 6.1. Let S be a fuzzy normal subgroup of a group G, and t E [0, 1]. Then St is a congruence relation on G.
Pro@
For any element a of G, s ( a a -1) ----#(e) = 1 _> t,
and so (a,a) E St. If (a,b) C St, then s(ab -I) > t. Since S is a fuzzy subgroup of G,
s(bo, -1) = s((ba-1)
= s(ab
> t,
which yields (b~a) E St. If (a,b) C St and (b,c) E ttt, then since S is a fuzzy subgroup of G,
s(ac -1) = #((ae)c -1) = s(a(b-lb)c -1) = s((ab-1)(bc-1)) >_ min{s(ab-1), s ( b c - 1 ) } > m i n { t , t } = t, and so (a,c) E St. Therefore St is an equivalence relation on G. To see t h a t S~ is a congruence relation on G, let (a, b) E St and z be any element of G. Then, since s(ab -1) > t,
s((az)(bz) -1) = # ( ( a z ) ( z - l b - 1 ) ) = s ( a ( z x - 1 ) b -1) = #(aeb -1) = s(ab -1) > t,
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and so (ax, bx) • #t. Since p is a fuzzy normal,
#((xa)(xb) -1) = # ( ( a x ) ( b - l x - 1 ) ) = p ( ( b - l x - 1 ) ( x a ) ) = p ( b - l ( x - l x ) a ) = #(b-lea) = p(b-la) = p ( a b -1) > t, and so (xa, xb) • #t. Therefore, Pt is a congruence relation on G. This completes the proof. [] Let # be a fuzzy subgroup of a group G, and #t be a t-level congruence relation of # on G. Let A be a nonempty subset of G. We denote by [x]u the congruence class of Pt containing the element x of G. Then the sets
#t (A) = {X • G: [x], C A}, #t^(A) = {X e G: [x], N A # 0}, are called, respectively, the lower and upper approximations of the set A with respect to #t. The following is due to Proposition 2.2 of [5]. We shall give a proof for completeness. PROPOSITION 6.2. Let # and A be fuzzy subgroups of a group G, and t • [0, 1]. Let A and B be nonempty subsets of G. Then thefoUowing hold:
(1) #t_ (A) c A c_ #t^(A), (2) #t^(A u B) = #t^(A) u #t^(B), (a) #t_ (A o B) = #t_ (A) n #t_ (B), (4) A c_ B implies #t_ (A) C_ #t_ (B), (5) A C_ B implies #,^(A) C_ #t^(B), (6) #t_ (A U B) D #t_ (A) U #t_ (B), n ,t^(B), (r) ,,t^(A n B) C_ (8) #t C At implies #t^(A C At^(A). Proof. (1) IfVa • #t_ (A) then a • [a]u C_A. Thus Pt (A) C A. Next, if a e A, then a E [a],. Thus, a • [a]u U A, and a • #t^(A). Thus, A c_ #t^(A).
(2) aE#~'(AUB)
~ [a],N(AUB)=0 ¢=> ([a]t, n A) U ([a]t, n B) = 0 4=> [ a ] ~ N A = ¢ or [ a ] ~ n B = 0 aEpt(A) or a ¢ # t ( A )
a E #t(A) U # t ( B ) . Thus, ttt(A U B) = #t^(A) U ltt^(B).
A P P R O X I M A T I O N S IN FUZZY G R O U P
215
(3) aE#t_(ANB)
~ [a], C_ANB ~=~ [a],C_A
and
¢=~ a E #t_ (A)
[a], C_B
and
a E #t_ (B)
A E #t_ (A) n #t_ (B). Thus, #t_ (A N B) = #t_ (A) N #t_ (B). (4) Since A C_ B, A O B = A. Then by (3), #t_(A) = # t _ ( A N B ) # L (A) OPt_ (B), which yields # L (A) c_ # L (B). (5) Since AC_B, A U B = B. Then by (4), pt^(A) = # t ^ ( A U B ) #t^(A) U #t^(B), which yields #t^(A) C pt^(B). (6) Since A C_ A U B and B _c A U B, by (4), we have pt_(A) c _ # t _ ( A u B )
and
= =
pt (B) C_pt_(AuB).
Thus, Pt_ (A) U #t_ (B) _C >t_ (A U B). (7) Since AN C_ A and AN C B, by (5), we have
#t^(mN t3) C_#t^(A)
and
#t^(A OB) C_#t^(B).
Thus, #t(A n B) C_#t^(A) N #t^(B). (8) If c E #t^(A), then 3x E [c], N A. Thus x E [ c ] , and x E A. Then (x, c) E #t C At, and so x E [c]a. Therefore, x E [c]~ n A, and c E ,kt^(A). Thus, #t^(A) C )~t^(A). This completes the proof. [] PROPOSITION 6.3. Let # be a fuzzy subgroup of a group G, and t E [0, 1]. If A and B are nonempty subsets of G, then
#t^(A)#t"(B) = #t^(AB). Proof. Let c be any element of #t^(A)pt^(B). Then c = ab with a E #t^(A) and b E #t^(B). Thus there exist elements x and y in G such that xE [a],nA
and
yE [b],nB.
And so x E [a],, x E A, y E [b]•, and y E B. Then, since #t is a congruence relation on G,
xy E [a],[b], = [ab], and
xy E AB.
This implies t h a t xy E [ab],OAB, and so c = ab E #t^(AB). Thus we have
#t"(A)#t^(S) C_,t^(AB).
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N. K U R O K I A N D P. P. W A N G
Conversely, let c be any element of ,at^(AB). T h e n there exists an element x in G such t h a t x E [ c ] , A AB. Thus x E [c], and x E AB. T h e n x = ab with a E A and b E B. Since c E [x], = [ab], = [a]~[b]~, we have c = yz with y = [a], and z = [b],, T h e n a E [y],, and so a E [y]~NA. T h u s y E ,at"(A). Similarly we have z E ,at^(B). Thus c = yz E #t^(A)pt"(B), and so ,at"(AB) C_ #t^(A),at^(B). Therefore, ,at^(AB) = #t^(A)pt^(B). This completes the proof. [] PROPOSITION 6,4. Let ,a be a fuzzy subgroup of a group G, and t E [0, 1]. If A and B are nonempty subsets of G, then ,at
(A),at_ (B) c_ ,at (AB).
Pro@ Let c be any element of ,at_ (A),at_ (B). T h e n c = ab with a E ,at(A) and b E ,at_(B). Thus [a], C A and [b], C B. Since ,at is a congruence relation on G, [c], = lab], = [a]u[b], C_ AB, and so c E ,at_ (AB). Thus, ,at_ (A),at_ (B) c_ pt_ (AB), which completes the proof. [] Let ,a and A be fuzzy subgroups of a group G. Then, as is well known and easily seen, , a n A is also a fuzzy subgroup G. LEMMA 6.5.
Let It and A be fuzzy subgroups of a group G, and t E [0, 1].
Then
(,a n A)t
= ,at N At.
Pro@ (a,b) E (# N A)t ¢~ (#N A)(ab -1) >_t ¢:> min{#(ab-1), A(ab-1)} >_ t
¢~ #(ab -1) >_ t ¢:> (a,b) E # t
and and
A(ab -1) >_ t (a,b) E A t
(a, b) ~ ,at n At. Therefore, (,a N A)t = ,at ~ At. This completes the proof.
[]
PROPOSITION 6.6. Let ,a and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a nonempty subset of G, then
(,a cl A)t^(A) = (,at n At)^(A) = ,at^(A) n At^(A).
APPROXIMATIONS IN FUZZY GROUP
217
Proof. Since #t and At are congruence relations on G, #t n At is also a congruence relation on G. It follows from Lemma 6.5 that (# n A)t^(A) = (pt n At)^(A), Vc E (p N A )t^( A ) ¢:~ [C],n.\ N A ¢ 0 3a E [c].n~ n A
¢:~ a E [c]..a E A aE[c]~NA ¢:~ c E #t"(A)
and and
and
a E [c]~.a E A aE[c]~nA
c ~ At"(A)
¢~ c e #,(A) n At^(A). Therefore, (# N A)t^(A) = #t^(A) N At^(A). This completes the proof.
[]
PROPOSITION 6.7. Let # and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a nonempty subset of G, then (# n A)t_ (A) = (#tAt)_ (A) = #t_ (A) N At_ (A). Proof.
It follows from Lemma 6.5 that (# n A)t_ (A) = (Pt n At)_ (A), VcE ( # n A ) t (A) ~
[c]~,n.~ C A
** [ c J t ~ A cE#t_(A)
and and
[c]~C_A cEAt_(A)
c E #t_ (A) n At_ (A). Therefore, (p [] A)t_ (A) = # t (A) rl At_ (A). This completes the proof.
[]
PROPOSITION 6.8. Let p be a fuzzy subgroup of a group G, and t E [0,1]. If A is a (normal) subgroup of G, then pt^(A) is a (normal) subgroup of G. Proof. Assume that A is a subgroup of G. Then, since e E [e]v and e E A, e E [e],NA. Thus, e E #t^(A). Let a and b be any elements of #t^(A). Then there exist elements x and y in G such that xE[a]~NA
and
y E [b]~nA.
Thus, x E [air, , y E [b]t,, and x, y E A. Since #t is a congruence relation on G, xy E [a]~[b]. = [ab]..
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N. K U R O K I A N D P. P. W A N G
Since A is a subgroup of G, xy E A. Thus, xy E [ab], n A, and ab E #t^( A ). Let a be any element of #t^(A). T h e n x E [a], n A for some x E G. Thus, x E [a], a n d x E A. S i n c e A i s a subgroup of G, x -1 E A. S i n c e # t is a congruence relation on G, (z, a) E gt
implies
(a -1, x -1) = ( a - l z x -1, a - l a x -1) E #t"
Thus, x -1 E [ a - l ] . . Therefore, x -1 E [ a - l ] , N A, and so a - 1 E #t^(A). This means t h a t #t"(A) is a subgroup of G. Assume t h a t A is a normal subgroup of G, Let a and x be any elements of #t^(A) and G, respectively. T h e n y E [a], c~ A for some y E G. Thus, y E [a], and y E A. Since A is normal,
x y x - 1 E x A x - 1 C A. Since #t is a congruence relation of G,
(y,a) E #t
implies
( x y x - Z , x a x -1) E #t.
Thus, x y x -1 E [xax-ll,. Therefore, xyx -1 E [xax-1], A A, and xax -1 E pt^(A). This means t h a t #t^(A) is a normal subgroup of G. [] PROPOSITION6.9. Let# be a fuzzy subgroup of a group G, and t E [0,1]. If A is a (normal) subgroup of G such that [e], C_ A, then #t_ (A) is a (normal) subgroup of G.
Proof. Assume t h a t A is a subgroup of G. Since (e, e) E #t, e E [e]t, C A. Let a and b be any elements of #t_ (A). T h e n [a], C_ A and [b]. c A. Since #t is a congruence on G, lab], = [a],[b], C_ A A C_ A. This implies t h a t ab E #t_(A). Let a be any element of # t _ ( A ) . T h e n [a], G A. Let x be any element of [a-Z],. T h e n (x,a -1) E #t, and so ( x - l , a ) E #t. Thus, x -1 E [a], _C A. Since A is a subgroup of G, x E A, and so [ a - 1 ] , C_ A. Thus, a -1 E #t_(A). This means t h a t # t _ ( A ) is a subgroup of G. Next assume t h a t A is normal. Let a and x be any element of #t (A) and G, respectively. T h e n [a], c_ A. Let z be any element of [xax-X],. T h e n (z, xax -1) E #t. Since #t is a congruence relation on G, ( x - l z x , a) E #t and so x - l z x E [a], C A. Thus, x - l z x = b for some b E a. Since A is normal, z = xbx -1 e x A x -1 c A, and so we have [xax-1], C_ A. Therefore, xax -1 E #t_ (A), which means t h a t #t_ (A) is normal. This completes the proof.
A P P R O X I M A T I O N S IN F U Z Z Y G R O U P LEMMA 6.10. Then
219
Let #, A be fuzzy subgroups of a group G, and t E [0, 1]. #t * At c_ (# o A)t,
where (*) is the product of binary relation on G and (o) is the product of fuzzy subsets of G. Proof. Let (a, b) be any element of #1 * At. T h e n there exists an element c of G such t h a t (a, c) E Pt and (c, b) E At. Therefore, we have #(ac -1) > t
and
A(cb -1) >_ t.
Then
(> o A )(ab -1) = sup[min{#(x), A(y)}: my = ab -1] _> min{p(ac-1), A(cb-1)} _> min{t, t} zt.
This implies t h a t (a, b) E (# o A)t, and so #t * At C (p o A)t. This completes the proof. [] REMARK 3. Let a and/3 be congruence relations on a group G. T h e n the product a * / 3 = {(a,b) E G x G: 3c E G, (a,c) E a and (c,b) E /3} is also a congruence relation on G. In fact, i f ( a , b ) E a * / 3 , t h e n 3 c E G, (a,c) E a a n d (c,b) E/3. T h e n (e,a-lc) E a, ( c - l , a -1) E a, ( a - l , c -1) E a, and (b, ac-lb) E a. And we have (b,c) E /3, ( b - l , c -1) E /3, (a, ac-lb) E /3, and (ac-lb, a) E /3. Thus, (b, a) E a */3. This means t h a t a */3 is symmetric, which implies t h a t a */3 is a congruence relation on G. PROPOSITION 6.11. Let # and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a nonempty subset of G, then
(~t * A~)^(A) C (p o A)F(A). Proof.
This follows from L e m m a 6.9 and (8) of Proposition 6.2.
[]
PROPOSITION 6.12. Let # and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a subgroup of G, then
pt^(A)flt^(A) c_ (#t * At)^(A).
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N. K U R O K I AND P. P. WANG
Proof. Let c be any element of # t ^ ( A ) A t ^ ( A ) . Then c = ab with a E # t " ( A ) and b E At"(A). Thus, there exist elements x and y in G such that x~ [a],NA
and
y E [b];~nA.
Therefore, z E [a],, y E [b]:~, and x , y E A. Since A is a subgroup of G, x y E A. Since #t and At are congruence relations on G, (x, a) E #t implies (xy, ay) E #t, and (y, b) E At implies (ay, ab) E At. Then we have (xy, c) = (xy, ab) E #t * At, and so x y E [c],t.:~t. Thus, x y E [c],~t.:,t N A, and so c E (#t * At)^(A). Therefore, we obtain that # t ^ ( A ) A t ^ ( A ) C (#t * At)^(A), which completes the proof. [] N. Kuroki would like to acknowledge the support of the Ministry of Education of Japan for providing financial assistance for the trip to the U.S. and the visit to Duke University and Creighton University in order to write this paper. He is also grateful to the Department of Electrical Engineering, Duke University, for providing facilities during the preparation of this work and to Professor S. C. Cheng ( Creighton University) for valuable comments and suggestions. The work of P. P. Wang was sponsored in part by the National Science Foundation under Grant No. ECS-9216~7~ and by the Electric Power Research Institute under" Grant No. RP8030-3. REFERENCES 1. 2.
3. 4. 5. 6. 7.
R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Math. 42:251-254 (1994). Z. Bonikowaski, Algebraic structures of rough sets, in W. P. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, Berlin, 1995, pp. 242-247. T. Iwinski, Algebraic approach to rough sets, Bull. Polish Acad. Sci. Math. 35:673-683 (1987). N. Kuroki, Fuzzy congruences and fuzzy normal subgroups, Inform. Sci. 60:247--259 (1992). Z. Pawlak, Rough sets, Int. J. Inf. Comp. Sci. 11:341-356 (1982). Z. Pawlak, Rough Sets Theoretical Aspects of Reasoning about Data, Kluwer Academic, Norwell, MA, 1991. J. Pomykala and J. A. Pomykala, The stone algebra of rough sets, Bull. Polish Acad. Sci. Math. 36:495-508 (1988).
Received 1 March 1995; revised 17 August 1995
NORTH-HOLLAND
The Lower and Upper
Approximations
in a Fuzzy Group
NOBUAKI KUROKI Department of Mathematics, Joetsu University of Education, Joetsu-shi, Niigata-kenn, Japan 943 and PAUL P. WANG Department of Electrical Engineering, Duke University, Box 90291, Durham, North Carolina 27708-0291
ABSTRACT' In this paper, we shall introduce the notion of a rough subgroup with respect to a normal subgroup of a group, and give some properties of the lower and the upper approximations in a group. Also, we will discuss a rough subgroup with respect to a t-level subset of a fuzzy normal subgroup.
1.
INTRODUCTION
T h e notion of rough sets was introduced by Pawlak in his paper [5]. Some authors, for example, Bonikowaski [2], Iwinski [3], and P o m y k a l a and P o m y k a l a [7], studied algebraic properties of rough sets. But there was no paper on rough sets in algebraic structures. Recently, Biswas and N a n d a [1] gave the notion of rough subgroups. Because their notion depends on the u p p e r approximation and does not depend on the lower approximation, we will discuss the lower and the upper approximation of a group and of a fuzzy group in detail. 2.
L O W E R A N D U P P E R A P P R O X I M A T I O N S IN A G R O U P
Let G be a group (as universe) with identity e. Let ~ be a congruence relation of G, t h a t is, $ is an equivalence relation on G such t h a t (a,b) e 6 ( a , b e G )
implies
(ax, bx) e 6
and
(xa, xb) e
for a l l x o f G . A subgroup N of G i s called normal if a N = N a for all a of G. Then, as is well known and easily seen, there exists a one-to-one
INFORMATION SCIENCES 90, 203-220 (1996) (~ Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0020-0255/96/$15.00 SSDI 0020-0255(95)00282-0
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N. K U R O K I AND P. P. WANG
correspondence between the set of all congruence relations on G and the set of all normal subgroups of G. Therefore, we can identify the notion of normal subgroups and of congruences on a group. Let N be a normal subgroup of G. Let A be a nonempty subset of G. Then the sets
N_(A) = { x • G : x N C A }
and
N^(A) = { x c G : x N n A ~ O }
are called, respectively, lower and upper approximations of a set A with respect to the normal subgroup N. Throughout this paper, H and N mean normal subgroups of G. The following is due to Proposition 2.2 of [5]. We shall give a proof for completeness. PROPOSITION 2.1. Let N and H be normal subgroups of a group G. Let A and B be any nonempty subsets of G. Then (1) (2) (3) (4) (5) (6) (7) (8)
N_(A) C A C N^(A). N^(A U B) = N^(A) U N^(B). N_ ( A n B) -- N_ (A) n N_ (B). A C_ B implies N_ (A) c_ N_ (B). A C_ B implies N^(A) C_N^(B). N_ (A u B) _~ N_ (A) U N_ (B). N^(A N B) C_ N^(A) n N^(B). N c_ H implies N"(A) C_H^(A).
Proof. (1) I f V a c N_(A), then a = ae e aN C_ A. T h u s N _ ( A ) C A . Next, ifVa E A, then a = ae C aN. Thus a E a N N A , that is, a N N A ~ O. This implies a E N^(A), and so A C N^(A). (2) aEN^(AUB)
¢:> a N N ( A U B ) ~ O tee ( a N N A ) U ( a N N B ) ~O ¢ez a N N A ~ O or a N N B ~ O ¢:> a E N^(A) or a c N^(B) a • N^(A) u N^(B).
Thus N^(A U B) = N^(A) U N^(B).
(3) a • N _ ( A N B ) ~ aN aN ¢=> a • ¢:> a E
C ANB C A and aN C_ B N_(A) and a • N _ ( B ) N_ (A) N N_ (B).
Thus N^(A n B) = N_ (A) N N_ (B).
A P P R O X I M A T I O N S IN FUZZY G R O U P
205
(4) Since A C_ B, A n B = A. Then by (3), we have N_ (A) = N= ( A n B) = N_ (A) N N_ (B), which yields N_ (A) C_ N_ (B). (5) Since A _c B, A U B = B. Then by (2), we have
N^(A) = N^(A U B) = N^(A) U N^(B), which yields N^(A) C_ N^(B). (6) A C A U B a n d B C _ A u B ; b y ( 4 ) , w e h a v e N_(A) C _ N _ ( A u B )
and
N_(B) CN_(AUB).
Thus N_(A) U N_(B) C_ N _ ( A U B). (7) A o B C_ A and A n B C_ B; by (g), we have
N^(A N 13) C_N^(A)
and
N^(A O B) C_N^(B).
Thus N^(A N B) C N"(A) N N^(B). (8) If Vc E N^(A), then 3x E cN n A, and so x E cN C_ cH. Therefore, x E cH n A. This implies c E H^(A), and N^(A) C H^(A). This completes the proof. [] PROPOSITION 2.2. Let N be a normal subgroup of a group G. Let A and B be nonempty subsets of G. Then
N^(A)N^(B) = N^(AB). Proof. Let c be any element of N^(AB). Then cN N A B ¢ O. Thus there exists an element x in G such that z E cN N A B , and so x E cN and zEAB. Thenz=abwithaEAandbE B. S i n c e c E x N = (ab),N= (aN)(bN), we have c = yz with y E aN and z E bN. Then a E yN, and so a E y N n A. Thus y E N^(A). Similarly we have z E N^(B). Thus c = yz E N^(A)N^(B), and so we have N^(AB) C_ N^(A)N^(B). Conversely, let c be any element of N^(A)N^(B), then c = ab with a E N^(A) and b E N^(B). Thus there exist elements x and y in G such that x E aN n A and y E bN N B, and so x E aN, x E A, y E bN, and y E B. Since N is normal, xy E (aN)(bN) = abN, and xy E AB. Thus xy E abN n AB, which yields that c = ab E N^(AB), and so N ^ ( A ) N ^ ( B ) C_ N"(AB). Therefore we have N^(A)N^(B) = N^(AB). This completes the proof. [] PROPOSITION 2.3. Let N be a normal subgroup of a group G. Let A and B be nonempty subsets of G. Then
N_ (A)N_ (B) g N_ (AB).
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N. KUROKI AND P. P. WANG
Proof. Let c be any element of N _ ( A ) N _ ( B ) . Then c = ab with a E N _ (A ) and b E N_(B). Thus aN C A a n d b N C_ B. S i n c e N i s n o r m a l , cN = abN = (aN)(bN) C_ AB, and so c E N _ ( A B ) . Thus N _ ( A ) N _ ( B ) C_ N_ (AB), which completes the proof. [] REMARK 1. Let H and N be normal subgroups of a group G. Then, as is well known and easily seen, H n N is also a normal subgroup of G. PROPOSITION 2.4. Let H and N be normal subgroups of a group G. Let A be a nonempty subset of G. Then
(H N N)^(A) = H^(A) n N^(A). Proof. VcE ( H N N ) ^ ( A ) ~ c ( H N N ) N A #O ¢* 3 a E c ( H N N ) N A aEc(HNN)
and
aEA
and
aEcN, aEA
¢=~ a E cH, a E A 3aEcHNA ~=~ c E H^(A)
and and
3aEcNNA cEN^(A)
c E H^(H) n N^(A). Thus we obtain that (H n N)^(A) = H^(A) n N^(A). This completes the proof. PROPOSITION 2.5. Let H and N be normal subgroups of a group G. Let A be a nonempty subset of G. Then (H n N ) _ (A) = H_ (A) n N_ (A).
Proo/. VcE ( H N N ) _ ( A ) ~=~ c ( H N N ) C A ~ cH C_ A ¢~ c E H _ ( A )
and
cN C_ A
and
cEN_(A)
¢~ c E H_ (A) n N_ (d). Thus we have ( H N N ) _ ( A ) -- H _ ( A ) n N_(A). proof.
This completes the []
A P P R O X I M A T I O N S IN F U Z Z Y G R O U P 3.
207
L O W E R A N D U P P E R R O U G H S U B G R O U P S IN A G R O U P
N ( A ) = ( N _ ( A ) , N ^ ( A ) ) is called a r o u g h set of A in G. A n o n e m p t y s u b s e t A of a g r o u p G is called an N ^ - r o u g h [normal] s u b g r o u p of G if t h e u p p e r a p p r o x i m a t i o n N ^ ( A ) of A is a [normal] s u b g r o u p of G. Similarly, a n o n e m p t y s u b s e t A of G is called an N _ - r o u g h [normal] s u b g r o u p of G if N _ (A) is a [normal] s u b g r o u p of G. PROPOSITION 3.1. Let N be a normal subgroup of a group G. If A is a subgroup of G, then it is an N^-rough subgroup of G.
Proof. Let e be t h e i d e n t i t y of G. Since N a n d A are s u b g r o u p s of G , e E A a n d e = ee E eN, a n d s o e E e N A A . T h u s e N N A ~ 9 . T h i s implies t h a t e E N ^ ( A ) . L e t a a n d b be any e l e m e n t s of N^(A). T h e n t h e r e exist e l e m e n t s x a n d y in G such t h a t x E a N N A a n d y E b N N A . T h u s x E aN, y E bN a n d x E A, y E A. Since A is a s u b g r o u p of G, xy E A. A n d since N is a n o r m a l s u b g r o u p of G, xy E (aN)(bN) = abN. T h u s xy E abN N A, a n d so ab E N " ( A ) . L e t a be a n y e l e m e n t of N^(A). T h e n x E a N A A for some x E G, t h a t is, x E a N a n d x E A. T h e n s i n c e A is a s u b g r o u p of G, x -1 E A. O n t h e o t h e r h a n d , since x = ah for some h E N , a n d since N is a n o r m a l s u b g r o u p of G, h -1 E N and we have x -1 = (ah) -1 = h - l a -1 E N a -1 = a - i N . T h u s x -1 E a - I N n A , a n d so a -1 E N^(A). T h e s e i m p l y t h a t N " ( A ) is a s u b g r o u p of G. This c o m p l e t e s t h e proof. [] PROPOSITION 3.2. Let N be a no~nal subgroup of a group G. If A is a normal subgroup of G, then it is an N^-rough normal subgroup of G.
Proof. It suffices to show t h a t N " ( A ) is normal. Let a a n d x be a n y e l e m e n t s of N ^ ( A ) a n d G, respectively. T h e n t h e r e exists an e l e m e n t y in G such t h a t y E a N A A, t h a t is, y E a N a n d y E A. Since N is n o r m a l , xyx -1 E
x ( a N ) x -1 = ( x a ) ( N x -1) = ( x a ) ( x - l N ) : ( x a x - 1 ) N .
Since A is n o r m a l ,
x y x - 1 E x A x - 1 C A. T h u s x y x -1 E ( x a x - 1 ) N N A, and so xax -1 E N^(A). N ^ ( A ) is n o r m a l , which c o m p l e t e s t h e proof.
This means that []
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PROPOSITION 3.3. Let N be a normal subgroup of a group G. If A is a subgroup of G such that N C A, then it is an N_-rough subgroup of G.
Proof. Since e N = N C A, e • N_ (A). Let a and b be any elements of N_ (A). Then a N C_ A and bN C_ A. Since N is a normal subgroup and A is a subgroup of G, abN = (aN)(bN) C_ A A C_ A. This implies that ab • N_ (A). Let a be any element of N_ (A). Then
a=ae•aNCA. Since A is a subgroup of G, a -1 • A. Thus we have
a - i N C A A C_ A. This implies that a -1 • N _ ( A ) . which completes the proof.
Therefore, N _ ( A ) is a subgroup of G, []
PROPOSITION 3.4. Let N be a normal subgroup of a group G. If A is a normal subgroup of G such that N C_ A, then it is an N_-rough normal subgroup of G.
Proof. It suffices to show that N_ (A) is normal. Let a and x be any element of N_ (A) and G, respectively. Then, a N C_ A. Since N and A are normal, ( x a x ) - l g = x ( a N ) x -1 C_ x A x -1 C_ A, and so xax -1 • N _ ( A ) , which means that N _ ( A ) is normal. This completes the proof. [] REMARK 2. The product H N of normal subgroups H and N of a group G is also a normal subgroup of G. So we can consider the lower and upper approximations ( H N ) _ (A) and ( H N ) ^ ( A ) for a nonempty subset A of G. PROPOSITION 3.5. Let H and N be normal subgroups of a group G. If A is a subgroup of G, then
H^(A)N^(A) C (HN)^(A). Proof. Let c be any element of H ^ ( A ) N ^ ( A ) . Then c = ab with a • HA(A) and b • N " ( A ) . Then there exist elements x and y in G such that x • a H A A and y • b N A A . T h u s x • aH, y • bN, x • A, and y • A. Then, since H is normal,
x y • (aH)(bN) = {a( H b ) N } = { a ( b H ) N } = { (ab)H } N = (ab)H N = cH N.
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Since A is a s u b g r o u p of G, xy E A. Therefore, xy E c H N N A, a n d so we have c E ( H N ) ^ ( A ) . T h u s H ^ ( A ) N ^ ( A ) C_ ( H N ) ^ ( A ) . T h i s c o m p l e t e s t h e proof. [] PROPOSITION 3.6. Let H and N be normal subgroups of a group G. If A is a subgroup of G, then
( H N ) ^ ( A ) C H ^ ( A ) N (1 N ^ ( A ) H . Proof. Let c be a n y element of ( H N ) ^ ( A ) . T h e n t h e r e exists an e l e m e n t x in G such t h a t x c c ( H N ) A A. T h u s x E c H N a n d x C A. T h u s x = cab w i t h a E H a n d b E N . Note t h a t a -1 E H a n d b -1 C N. Since H is n o r m a l , x = cab E crib = cbH, a n d so x E cbH N A. T h u s cb E H ^ ( A ) , a n d so c c H^(A)b -1 c_ H ^ ( A ) N . Similarly, c E N ^ ( A ) H . Therefore, c e H ^ ( A ) N A N ^ ( A ) H , and so ( H N ) ^ ( A ) C_ H ^ ( A ) N A N ^ ( A ) H , which c o m p l e t e s t h e proof. [] PROPOSITION 3.7. Let H and N be normal subgroups of a group G. If A is a subgroup of G, then
H _ ( A ) N _ ( A ) c_ ( H N ) _ ( A ) . Proof. Let c be a n y element of H _ ( A ) N _ ( A ) . T h e n c = ab w i t h a E H _ (A) a n d b E N _ (A). T h u s a H C_ A a n d bN C_ A. Since H is a n o r m a l s u b g r o u p a n d A is a s u b g r o u p of G, we have ( a b ) H N = { a ( b H ) } N = { a ( H b ) } N = { ( a H ) b } N = (aH)(bN) C_ A A C_ A. Therefore, we have c = ab e ( H N ) _ (A), a n d so H_ (A)N_ (A) C_( H N ) _ (A), which c o m p l e t e s t h e proof. []
4.
R O U G H S U B G R O U P S IN A F A C T O R G R O U P
T h e lower a n d u p p e r a p p r o x i m a t i o n s can be p r e s e n t e d in a n equivalent form as shown below. Let N be a n o r m a l s u b g r o u p of a g r o u p G, a n d A a n o n e m p t y s u b s e t of G. T h e n
N_(A) = {aN e G/N:
a N C A},
N^(A)={aNeG/N:
aNnA~O}.
PROPOSITION 4.1. Let N be a normal subgroup of a group G. If A is a subgroup of G, then N ^ ( A ) is a subgroup of G / N .
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Proof. N o t e t h a t N = e N is t h e i d e n t i t y of G / N . Since e E N = e N a n d e E A , e E e N C ~ A . T h u s N = e N E N ^ ( A ) . Let a N a n d bN b e a n y e l e m e n t s of N ^ ( A ) . T h e n t h e r e exist e l e m e n t s x a n d y in G such t h a t x E aNAA and y E b N n A . T h u s x E a N , y E bN, x E A, a n d y E A. T h e n , since N is normal, x y E ( a N ) ( b N ) = (ab)N. Since A is a s u b g r o u p of G, x y E A. T h u s x y E (ab)N A A. T h e r e f o r e we have ( a N ) ( b N ) = (ab)N E N ^ ( A ) . Let a N be a n y e l e m e n t of N ^ ( A ) . T h e n t h e r e exists an e l e m e n t x in G such t h a t x E a N A A. T h u s x E a N a n d x E A. T h u s x = an for some n E N . Note t h a t ( a N ) -1 = a - I N . T h e n we have x -1 = (an) -1 = n - l a -1 E N a -1 = a - i N . Since x -1 E A, x -1 E a - i N N A, a n d so a - i N E N ^ ( A ) . Therefore, we o b t a i n t h a t N ^ ( A ) is a s u b g r o u p of G / N , which c o m p l e t e s t h e proof. [] PROPOSITION 4.2. Let N be a normal subgroup of a group G. If A is a normal subgroup of G, then N ^ ( A ) is a normal subgroup of G / N .
Proof. Let a N a n d x N be a n y e l e m e n t s of N ^ ( A ) a n d G / N , respectively. T h e n t h e r e exists an e l e m e n t y in G such t h a t y E a N A A. T h u s y E a N a n d y E A. Since N is normal, we have x y x -1 E x ( a N ) x -1 = ( x a x - 1 ) N . A n d since A is also normal,
x y x - 1 E x A x - 1 C_ A. T h u s x y x -1 E ( x a x - 1 ) N N A, and so
( x N ) ( a N ) ( x N ) -1 = ( x N ) ( a N ) ( x - l g ) = ( x a x - 1 ) N E N ^ ( A ) . T h e n it follows from this and P r o p o s i t i o n 4.1 t h a t N ^ ( A ) is a n o r m a l s u b g r o u p of G I N . T h i s c o m p l e t e s t h e proof. [] PROPOSITION 4.3. Let N be a normal subgroup of a group G. I f A is a normal subgroup of G such that N C A, then N_ (A) is a subgroup of G/N.
Proof. Since e N = N C A, e N E N_ (A). Let a N a n d bN b e a n y e l e m e n t s of N _ ( A ) ; t h e n a N C A and bN C A. Since N is n o r m a l , abN = ( a N ) ( b N ) C A A C_ A,
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and so we have
( a N ) ( b N ) = ab(N) C N _ ( A ) . Let a N be any element of N_(A). Then, since a = ae • a N C A, a -1 E A. Thus
a - i N C_ A A C_ A. This means that
(aN) -1 = a - i N C N _ ( A ) . Thus N_ (A) is a subgroup of G / N , which completes the proof.
[]
PROPOSITION 4.4." Let N be a normal subgroup of a group G. If A is a normal subgroup of G such that N C_ A, then N_ (A) is a normal subgroup
ola/N. Proof. Let a N and x N be any elements of N _ ( A ) and G / N , respectively. Then a N C_ A. Since N and A are normal, ( x a x - 1 ) N = ( x a ) ( x - l N ) = ( x a ) ( N x -~) = x ( a N ) x -~ C_ x A x -1 C A. This implies that
( x N ) ( a N ) ( x N ) -1 = ( x N ) ( a N ) ( x - l N )
= (xax-1)N E N_(A).
Then it follows from this and Proposition 4.3 that N _ ( A ) is a normal subgroup of G / N . This completes the proof. []
5.
H O M O M O R P H I C IMAGES OF THE U P P E R A P P R O X I M A T I O N
Let G and G ~ be two groups. A mapping f: G --~ G ~ is called a homomorphism from G to G t if
f(ab) = f ( a ) f ( b ) for all a, b E G. We denote by e ~ the identity of G( Then the set Ker(f) = {x • G: f ( x ) = e'} is called the Kernel of f. group of G.
As is well known, Ker(f) is a normal sub-
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LEMMA 5.1. Let f be a homornorphism of a gwup G to a group G'. If A is a nonempty subset of G, then
f(A) ~ f(A Ker(f)). Proof. L e t y be any element of f ( A ) . T h e n f(a) = y for s o m e a E A. W e n o t e t h a t e E K e r ( f ) . T h u s we have y = f(a) = f(ae) E f ( A K e r ( f ) ) , a n d so f ( A ) c_ f ( A K e r ( f ) ) . Conversely, let y be a n y e l e m e n t of f ( A K e r ( f ) ) . T h e n f(a) = y for some a E A K e r ( f ) . T h u s a = b c w i t h b E A a n d c ¢ Ker(f). Then
y = f(a) = f(bc) = f(b)f(c) = f(b)e I = f(b) E f ( A ) , a n d so f ( A K e r ( f ) ) C_ f ( A ) . Therefore, f ( A ) = f ( A K e r ( f ) ) . PROPOSITION 5.2. Let f be a homomorphism of G to G r, and N a normal subgroup of G. If A is a nonempty subset of G, then
f ( A ) c_ f ( N ^ ( A ) ) c_ f ( A N ) . Proof. B y P r o p o s i t i o n 2.1(1), A C_ N^(A), a n d since f is a h o m o m o r p h i s m of G t o G ' , we have f ( A ) C f ( N ^ ( A ) ) . To see I ( N ^ ( A ) ) C f ( A N ) , let y be a n y e l e m e n t of f ( N ^ ( A ) ) ; t h e n f(a) = y for some a E N^(A). T h u s t h e r e exists an e l e m e n t x in G such t h a t x E aN N A. T h e n x E a N a n d x E A. T h u s x = ab for some b E N , t h a t is, a = xb -1. Since N is a s u b g r o u p of G, b - i E N . T h e n we have y = f(a) = f(xb -1) E f ( A N ) , []
a n d so f ( N ^ ( A ) ) c_ f ( A N ) , which c o m p l e t e s t h e proof.
PROPOSITION 5.3. Let f be a homomorphism of G to G I. If A is a nonernpty subset of G, then
f(A) = f(Ker(f)^(A)). Pro@ Since K e r ( f ) is a n o r m a l s u b g r o u p of G, L e m m a 5.1 a n d P r o p o s i t i o n 5.2 t h a t
it follows from
f ( A ) c_ f ( K e r ( f ) ^ ( A ) ) c_ f ( A K e r ( f ) ) = f ( A ) . Therefore, we o b t a i n t h a t f ( A ) = f ( K e r ( f ) ^ ( A ) ) , which c o m p l e t e s t h e proof.
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6. T H E L O W E R A N D U P P E R A P P R O X I M A T I O N S ON A F U Z Z Y NORMAL SUBGROUP A fuzzy subset # of a group G is called a fuzzy subgroup of G if the following conditions hold: (1) #(ab) > min{#(a),#(b)} for all a,b ~ G, (2) , ( a -1) _> s(a) for all a E G, (3) s ( e ) = 1. A fuzzy s u b g r o u p S of G is called a fuzzy subgroup of G if (4) s(ab) = s(ba) for all a. b E G. Let S be a fuzzy normal subgroup of G. For each t E [0, 1], the set St = { ( a , b ) E G x G : s ( a b
-1) >_t}
is called a t-level relation of S. T h e n we have the following. LEMMA 6.1. Let S be a fuzzy normal subgroup of a group G, and t E [0, 1]. Then St is a congruence relation on G.
Pro@
For any element a of G, s ( a a -1) ----#(e) = 1 _> t,
and so (a,a) E St. If (a,b) C St, then s(ab -I) > t. Since S is a fuzzy subgroup of G,
s(bo, -1) = s((ba-1)
= s(ab
> t,
which yields (b~a) E St. If (a,b) C St and (b,c) E ttt, then since S is a fuzzy subgroup of G,
s(ac -1) = #((ae)c -1) = s(a(b-lb)c -1) = s((ab-1)(bc-1)) >_ min{s(ab-1), s ( b c - 1 ) } > m i n { t , t } = t, and so (a,c) E St. Therefore St is an equivalence relation on G. To see t h a t S~ is a congruence relation on G, let (a, b) E St and z be any element of G. Then, since s(ab -1) > t,
s((az)(bz) -1) = # ( ( a z ) ( z - l b - 1 ) ) = s ( a ( z x - 1 ) b -1) = #(aeb -1) = s(ab -1) > t,
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and so (ax, bx) • #t. Since p is a fuzzy normal,
#((xa)(xb) -1) = # ( ( a x ) ( b - l x - 1 ) ) = p ( ( b - l x - 1 ) ( x a ) ) = p ( b - l ( x - l x ) a ) = #(b-lea) = p(b-la) = p ( a b -1) > t, and so (xa, xb) • #t. Therefore, Pt is a congruence relation on G. This completes the proof. [] Let # be a fuzzy subgroup of a group G, and #t be a t-level congruence relation of # on G. Let A be a nonempty subset of G. We denote by [x]u the congruence class of Pt containing the element x of G. Then the sets
#t (A) = {X • G: [x], C A}, #t^(A) = {X e G: [x], N A # 0}, are called, respectively, the lower and upper approximations of the set A with respect to #t. The following is due to Proposition 2.2 of [5]. We shall give a proof for completeness. PROPOSITION 6.2. Let # and A be fuzzy subgroups of a group G, and t • [0, 1]. Let A and B be nonempty subsets of G. Then thefoUowing hold:
(1) #t_ (A) c A c_ #t^(A), (2) #t^(A u B) = #t^(A) u #t^(B), (a) #t_ (A o B) = #t_ (A) n #t_ (B), (4) A c_ B implies #t_ (A) C_ #t_ (B), (5) A C_ B implies #,^(A) C_ #t^(B), (6) #t_ (A U B) D #t_ (A) U #t_ (B), n ,t^(B), (r) ,,t^(A n B) C_ (8) #t C At implies #t^(A C At^(A). Proof. (1) IfVa • #t_ (A) then a • [a]u C_A. Thus Pt (A) C A. Next, if a e A, then a E [a],. Thus, a • [a]u U A, and a • #t^(A). Thus, A c_ #t^(A).
(2) aE#~'(AUB)
~ [a],N(AUB)=0 ¢=> ([a]t, n A) U ([a]t, n B) = 0 4=> [ a ] ~ N A = ¢ or [ a ] ~ n B = 0 aEpt(A) or a ¢ # t ( A )
a E #t(A) U # t ( B ) . Thus, ttt(A U B) = #t^(A) U ltt^(B).
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(3) aE#t_(ANB)
~ [a], C_ANB ~=~ [a],C_A
and
¢=~ a E #t_ (A)
[a], C_B
and
a E #t_ (B)
A E #t_ (A) n #t_ (B). Thus, #t_ (A N B) = #t_ (A) N #t_ (B). (4) Since A C_ B, A O B = A. Then by (3), #t_(A) = # t _ ( A N B ) # L (A) OPt_ (B), which yields # L (A) c_ # L (B). (5) Since AC_B, A U B = B. Then by (4), pt^(A) = # t ^ ( A U B ) #t^(A) U #t^(B), which yields #t^(A) C pt^(B). (6) Since A C_ A U B and B _c A U B, by (4), we have pt_(A) c _ # t _ ( A u B )
and
= =
pt (B) C_pt_(AuB).
Thus, Pt_ (A) U #t_ (B) _C >t_ (A U B). (7) Since AN C_ A and AN C B, by (5), we have
#t^(mN t3) C_#t^(A)
and
#t^(A OB) C_#t^(B).
Thus, #t(A n B) C_#t^(A) N #t^(B). (8) If c E #t^(A), then 3x E [c], N A. Thus x E [ c ] , and x E A. Then (x, c) E #t C At, and so x E [c]a. Therefore, x E [c]~ n A, and c E ,kt^(A). Thus, #t^(A) C )~t^(A). This completes the proof. [] PROPOSITION 6.3. Let # be a fuzzy subgroup of a group G, and t E [0, 1]. If A and B are nonempty subsets of G, then
#t^(A)#t"(B) = #t^(AB). Proof. Let c be any element of #t^(A)pt^(B). Then c = ab with a E #t^(A) and b E #t^(B). Thus there exist elements x and y in G such that xE [a],nA
and
yE [b],nB.
And so x E [a],, x E A, y E [b]•, and y E B. Then, since #t is a congruence relation on G,
xy E [a],[b], = [ab], and
xy E AB.
This implies t h a t xy E [ab],OAB, and so c = ab E #t^(AB). Thus we have
#t"(A)#t^(S) C_,t^(AB).
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Conversely, let c be any element of ,at^(AB). T h e n there exists an element x in G such t h a t x E [ c ] , A AB. Thus x E [c], and x E AB. T h e n x = ab with a E A and b E B. Since c E [x], = [ab], = [a]~[b]~, we have c = yz with y = [a], and z = [b],, T h e n a E [y],, and so a E [y]~NA. T h u s y E ,at"(A). Similarly we have z E ,at^(B). Thus c = yz E #t^(A)pt"(B), and so ,at"(AB) C_ #t^(A),at^(B). Therefore, ,at^(AB) = #t^(A)pt^(B). This completes the proof. [] PROPOSITION 6,4. Let ,a be a fuzzy subgroup of a group G, and t E [0, 1]. If A and B are nonempty subsets of G, then ,at
(A),at_ (B) c_ ,at (AB).
Pro@ Let c be any element of ,at_ (A),at_ (B). T h e n c = ab with a E ,at(A) and b E ,at_(B). Thus [a], C A and [b], C B. Since ,at is a congruence relation on G, [c], = lab], = [a]u[b], C_ AB, and so c E ,at_ (AB). Thus, ,at_ (A),at_ (B) c_ pt_ (AB), which completes the proof. [] Let ,a and A be fuzzy subgroups of a group G. Then, as is well known and easily seen, , a n A is also a fuzzy subgroup G. LEMMA 6.5.
Let It and A be fuzzy subgroups of a group G, and t E [0, 1].
Then
(,a n A)t
= ,at N At.
Pro@ (a,b) E (# N A)t ¢~ (#N A)(ab -1) >_t ¢:> min{#(ab-1), A(ab-1)} >_ t
¢~ #(ab -1) >_ t ¢:> (a,b) E # t
and and
A(ab -1) >_ t (a,b) E A t
(a, b) ~ ,at n At. Therefore, (,a N A)t = ,at ~ At. This completes the proof.
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PROPOSITION 6.6. Let ,a and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a nonempty subset of G, then
(,a cl A)t^(A) = (,at n At)^(A) = ,at^(A) n At^(A).
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Proof. Since #t and At are congruence relations on G, #t n At is also a congruence relation on G. It follows from Lemma 6.5 that (# n A)t^(A) = (pt n At)^(A), Vc E (p N A )t^( A ) ¢:~ [C],n.\ N A ¢ 0 3a E [c].n~ n A
¢:~ a E [c]..a E A aE[c]~NA ¢:~ c E #t"(A)
and and
and
a E [c]~.a E A aE[c]~nA
c ~ At"(A)
¢~ c e #,(A) n At^(A). Therefore, (# N A)t^(A) = #t^(A) N At^(A). This completes the proof.
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PROPOSITION 6.7. Let # and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a nonempty subset of G, then (# n A)t_ (A) = (#tAt)_ (A) = #t_ (A) N At_ (A). Proof.
It follows from Lemma 6.5 that (# n A)t_ (A) = (Pt n At)_ (A), VcE ( # n A ) t (A) ~
[c]~,n.~ C A
** [ c J t ~ A cE#t_(A)
and and
[c]~C_A cEAt_(A)
c E #t_ (A) n At_ (A). Therefore, (p [] A)t_ (A) = # t (A) rl At_ (A). This completes the proof.
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PROPOSITION 6.8. Let p be a fuzzy subgroup of a group G, and t E [0,1]. If A is a (normal) subgroup of G, then pt^(A) is a (normal) subgroup of G. Proof. Assume that A is a subgroup of G. Then, since e E [e]v and e E A, e E [e],NA. Thus, e E #t^(A). Let a and b be any elements of #t^(A). Then there exist elements x and y in G such that xE[a]~NA
and
y E [b]~nA.
Thus, x E [air, , y E [b]t,, and x, y E A. Since #t is a congruence relation on G, xy E [a]~[b]. = [ab]..
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N. K U R O K I A N D P. P. W A N G
Since A is a subgroup of G, xy E A. Thus, xy E [ab], n A, and ab E #t^( A ). Let a be any element of #t^(A). T h e n x E [a], n A for some x E G. Thus, x E [a], a n d x E A. S i n c e A i s a subgroup of G, x -1 E A. S i n c e # t is a congruence relation on G, (z, a) E gt
implies
(a -1, x -1) = ( a - l z x -1, a - l a x -1) E #t"
Thus, x -1 E [ a - l ] . . Therefore, x -1 E [ a - l ] , N A, and so a - 1 E #t^(A). This means t h a t #t"(A) is a subgroup of G. Assume t h a t A is a normal subgroup of G, Let a and x be any elements of #t^(A) and G, respectively. T h e n y E [a], c~ A for some y E G. Thus, y E [a], and y E A. Since A is normal,
x y x - 1 E x A x - 1 C A. Since #t is a congruence relation of G,
(y,a) E #t
implies
( x y x - Z , x a x -1) E #t.
Thus, x y x -1 E [xax-ll,. Therefore, xyx -1 E [xax-1], A A, and xax -1 E pt^(A). This means t h a t #t^(A) is a normal subgroup of G. [] PROPOSITION6.9. Let# be a fuzzy subgroup of a group G, and t E [0,1]. If A is a (normal) subgroup of G such that [e], C_ A, then #t_ (A) is a (normal) subgroup of G.
Proof. Assume t h a t A is a subgroup of G. Since (e, e) E #t, e E [e]t, C A. Let a and b be any elements of #t_ (A). T h e n [a], C_ A and [b]. c A. Since #t is a congruence on G, lab], = [a],[b], C_ A A C_ A. This implies t h a t ab E #t_(A). Let a be any element of # t _ ( A ) . T h e n [a], G A. Let x be any element of [a-Z],. T h e n (x,a -1) E #t, and so ( x - l , a ) E #t. Thus, x -1 E [a], _C A. Since A is a subgroup of G, x E A, and so [ a - 1 ] , C_ A. Thus, a -1 E #t_(A). This means t h a t # t _ ( A ) is a subgroup of G. Next assume t h a t A is normal. Let a and x be any element of #t (A) and G, respectively. T h e n [a], c_ A. Let z be any element of [xax-X],. T h e n (z, xax -1) E #t. Since #t is a congruence relation on G, ( x - l z x , a) E #t and so x - l z x E [a], C A. Thus, x - l z x = b for some b E a. Since A is normal, z = xbx -1 e x A x -1 c A, and so we have [xax-1], C_ A. Therefore, xax -1 E #t_ (A), which means t h a t #t_ (A) is normal. This completes the proof.
A P P R O X I M A T I O N S IN F U Z Z Y G R O U P LEMMA 6.10. Then
219
Let #, A be fuzzy subgroups of a group G, and t E [0, 1]. #t * At c_ (# o A)t,
where (*) is the product of binary relation on G and (o) is the product of fuzzy subsets of G. Proof. Let (a, b) be any element of #1 * At. T h e n there exists an element c of G such t h a t (a, c) E Pt and (c, b) E At. Therefore, we have #(ac -1) > t
and
A(cb -1) >_ t.
Then
(> o A )(ab -1) = sup[min{#(x), A(y)}: my = ab -1] _> min{p(ac-1), A(cb-1)} _> min{t, t} zt.
This implies t h a t (a, b) E (# o A)t, and so #t * At C (p o A)t. This completes the proof. [] REMARK 3. Let a and/3 be congruence relations on a group G. T h e n the product a * / 3 = {(a,b) E G x G: 3c E G, (a,c) E a and (c,b) E /3} is also a congruence relation on G. In fact, i f ( a , b ) E a * / 3 , t h e n 3 c E G, (a,c) E a a n d (c,b) E/3. T h e n (e,a-lc) E a, ( c - l , a -1) E a, ( a - l , c -1) E a, and (b, ac-lb) E a. And we have (b,c) E /3, ( b - l , c -1) E /3, (a, ac-lb) E /3, and (ac-lb, a) E /3. Thus, (b, a) E a */3. This means t h a t a */3 is symmetric, which implies t h a t a */3 is a congruence relation on G. PROPOSITION 6.11. Let # and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a nonempty subset of G, then
(~t * A~)^(A) C (p o A)F(A). Proof.
This follows from L e m m a 6.9 and (8) of Proposition 6.2.
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PROPOSITION 6.12. Let # and A be fuzzy subgroups of a group G, and t E [0, 1]. If A is a subgroup of G, then
pt^(A)flt^(A) c_ (#t * At)^(A).
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N. K U R O K I AND P. P. WANG
Proof. Let c be any element of # t ^ ( A ) A t ^ ( A ) . Then c = ab with a E # t " ( A ) and b E At"(A). Thus, there exist elements x and y in G such that x~ [a],NA
and
y E [b];~nA.
Therefore, z E [a],, y E [b]:~, and x , y E A. Since A is a subgroup of G, x y E A. Since #t and At are congruence relations on G, (x, a) E #t implies (xy, ay) E #t, and (y, b) E At implies (ay, ab) E At. Then we have (xy, c) = (xy, ab) E #t * At, and so x y E [c],t.:~t. Thus, x y E [c],~t.:,t N A, and so c E (#t * At)^(A). Therefore, we obtain that # t ^ ( A ) A t ^ ( A ) C (#t * At)^(A), which completes the proof. [] N. Kuroki would like to acknowledge the support of the Ministry of Education of Japan for providing financial assistance for the trip to the U.S. and the visit to Duke University and Creighton University in order to write this paper. He is also grateful to the Department of Electrical Engineering, Duke University, for providing facilities during the preparation of this work and to Professor S. C. Cheng ( Creighton University) for valuable comments and suggestions. The work of P. P. Wang was sponsored in part by the National Science Foundation under Grant No. ECS-9216~7~ and by the Electric Power Research Institute under" Grant No. RP8030-3. REFERENCES 1. 2.
3. 4. 5. 6. 7.
R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Math. 42:251-254 (1994). Z. Bonikowaski, Algebraic structures of rough sets, in W. P. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, Berlin, 1995, pp. 242-247. T. Iwinski, Algebraic approach to rough sets, Bull. Polish Acad. Sci. Math. 35:673-683 (1987). N. Kuroki, Fuzzy congruences and fuzzy normal subgroups, Inform. Sci. 60:247--259 (1992). Z. Pawlak, Rough sets, Int. J. Inf. Comp. Sci. 11:341-356 (1982). Z. Pawlak, Rough Sets Theoretical Aspects of Reasoning about Data, Kluwer Academic, Norwell, MA, 1991. J. Pomykala and J. A. Pomykala, The stone algebra of rough sets, Bull. Polish Acad. Sci. Math. 36:495-508 (1988).
Received 1 March 1995; revised 17 August 1995