In this pap r we introduce a method to expa d the category 1of all finite-dimensional fuzzy spaces associated with finie-dime sional Chu spaces in to a complete system... In fact, we hav
Trang 1T? - p chi Tin hoc v fi'eu khi€n hqc, T.17, S.2 (2001), 35 - 38
FUZZY SPACES
NGUYEN NHUY, PHAM QUANG TRINH
and
VU THI HONG THANH
Abstract In this pap r we introduce a method to expa d the category 1of all finite-dimensional fuzzy spaces associated with finie-dime sional Chu spaces in to a complete system
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I is sh ws in [ 7 that, the category 1of all finite-dime sio al fuzzy spaces associated with finite-dimensio al Ch spaces is an equivale t system Unforunately, 1 is n t closed under the cross product, th refore 1 is n t a complete system In this p p r we introduce a method to expan th
ctegory 1into a c mplete system, that is, we constru t a "dual" n-set category 1 *containing 1as
a su category, where 1 *is a complete system
By n - s et we mean a cartesian product X = 11 ;~ 1Xi Let S denote th n-set category, when the category S* is defn d as folows:
1 Objects of S* are morphisms in S
2 If a : X = 1 ; ~1X; - t Y = 11; ~ 1Y ; a d a ' :
objects of S*, the a morphism <p : a - t a' from a to
ip : Y = 11 7=1Y i - t X ' = 1 17 = 1X:
Let a : X = 11nt=1 X·t -t Y = 11t=ln y:t a' : X' = 111n = 1 X't - t Y' = 11nt = 1 Y't and a" • X" =
11 ;~ 1X:' - t Y " = 11 ;~1y';" be objects in S*, < p : a - t a ' a d < ' : a' - t a" be morphisms of S* (i.e
ip : Y = 117 = 1Y; - t X' = 1 17=1X: and <p ' : Y ' = 1 17=1Y/ - t X" = 11; ~ 1X : ' ).
The c ompo s i ion of < p and i p" , denoted by < p'*< ,is give by
a ' in S* is a map (n the n-set category)
< '* < = < 'a' < : a - t a "
It is easy to chek that with th above d fniton S* is a category
For a give set X =1 1; '=1Xi, let X* =[ 0 , l x d note collection of all fuzzy sets of X For a ma
a : X = 11 ;~1Xi - t Y = 11 ;~1Y; we d fn the conjugate a * : Y * - t X * of a by the formula
a * (a )( x) = a(a(x)) for x EX a d a E Y
It is easy to see that
( Bar = a * B * for e ery a : X Y an B : Y Z.
Trang 2Now for a :X = r = l Xi - t Y = TI ~ '= 1Y ; we define F * ( a ) = (TI 7 = 1Xi , f a , Y * ), where Y * denotes
the collection of all fuzzy s ts of Y = TI ~ 1 y ; , and f a :TI 7 = 1 Xi X Y * - t [0,1] isgiven by
f a (Xl , X2, · , X n, a ) = a (a( x I ' X 2 , , x n)) for every (Xl , X2, , Xn, a) ETI7 =1 Xi X Y *
The (n+1)-dimensional Chu space F * ( a ) = (TI 7 = 1X i, f a , Y * ) is called the (n + l ) -dim e n s ional
* -fu zz y s pac e a ss o c i a t ed w ith th e m ap a : X = TI 7= 1 Xi - t Y = TI ~ 1 Y; The category of all (n+1)-dimensio al *-fuzzy spaces associated with maps in the n-set c tegory S is called the (n +
3 RE SU LT S
At first we will sh w that the (n + 1)-dimensional *-fuzzy category 1 * defined above contains
the c tegory 1a a subcategory In fact, we have the following theorem
*-fuzzy space
\
Theorem 2 1 * i s a compl e te s s tem.
Proof Assume that <I> = (TI7 =1< Pi,1 f; ) : F * ( a ) = (TI 7 = 1X i f a , Y * ) - t F * ( a = (TI 7 ~ 1X: , f a', Y' * )
is a (n+1)-Chu morphism, where F * ( a ) and F * ( a are (n+1)-dimensional *-fuzzy spaces associated
with the maps a = TI~ ' = 1cc ; :X = TI: ~1 X i - t Y = TI 7 = 1Y; and a ' = TI~ ' = 1a ; :X' = TI 7 = 1 X; - t
Y ' = TI n i = l iY '' respectiv.e 1y. Putt m g (3= a,<P= TI n i = l a i <P, i : X = TIn X i = l i - t Y' = TIn i = l i'Y' we
get the cross product C = (TI7 =1X i, f a X < I> f a ' Y' * ), which is a (n+1)-dimensional *-fuzzy space
associated with the map (3= TI7 =I a ; < Pi ·In fact, for every ( XI , . , Xn, b ) E TI 7=I X i X Y ' *, we have
(t o X 'I' f a ' )(XI,' " , Xn, b ) =f a ' ( < pdxd, ··· , < P n (xn), b)
= b( a < p I(xd, ··· ,a ~ < P n ( X n))
= f a ' P (xI ,' " , xn,b )
=f { 1 (xI,'" ,xn,b).
Th s, the category 1* is closed under the cross product Therefore the theorem isproved
Theorem 3 F * S - t 1 * i s a c ov ariant functor
n
F * ( < p) = (II< Pi a < p a '* )
i= l
where ip" and a' * are conjugated of< P=TI7 = 1< P i and a' =TI 7 = 1a ; , respectively, that is
n
<p* ( a )( YI"" , Y n ) = a ( < pdyd ,· · , < Pn ( Yn )) for every ( Yl, '" ,Yn ) E II Y ; and a EX' *
i= 1
a/ * (b)( x ~ , ,x~) = b( a ~ (x~), , a ~(x~)) for every (x~, , x~) E IIX: and bE y' *
i = l
We claim that F * ( < p ) : F * ( a ) = (TI 7= IX,f o, Y * ) - t F * ( a = (TI 7= IX: , a "Y' * ) is a (n +
1)-dimensio al Chu morphism That is, the following diagram commutes:
Trang 3COMPLETION OF THE CATEGORY OF FINITE-DIMENSIONAL FUZZY SPACES 37
[[' ; = 1 X i X v ':
- - >1 i= 1 i X Y *
f a ( x 1, '" , x n, < p a/ * ( b )) = < p a' * (b)( a dx1) , " ,an ( xn ))
= ( a ' < p) * (b)( a dxd, ·· · , a n( xn ))
=b( a ~ < P1 a dxd , ,a~ < Pnan ( xn ))
= fa' (<pa(x) , b)
Hence F * (<p) = (IT7=1 <Piai, <p * a' * ) is a (n+1)-Chu morphism
a = i = 1 ai - > a = i= 1 ai e morp Isms In le., < p= i= 1 < Pi : = i = ; 1 i - > =
rr = X: and < p '= IT 7 1< p ; Y' = IT 7 1Y - > X" = IT 7=1 X:' are maps in the n-set category) By
the definition we have < p'* < p= < p ' a ' < p= IT 7 1< p ~ a : < pi' Therefore
F * (' < p *< p ) = (' < paI s p ec, (' < paI < p )*a" *)
= < pa < pa, < pa < p a
= F * ( < p')F * ( < p)
E l ec tr o ic No t es in Th eo r est i cal Co mput e Scie n ce , 1 7 9.
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Rec e ived Augu s t 11, 2000
D e p r tme nt of I n formation Technology,
Vin h U n iv er s i y , Nqh e An , Vietnam