D:\User\sbornik_38_pdf\21.DVI


Mathematical Problems of Computer Science 38, 49{52, 2012.

Algebr as with Fuzzy Oper ations

S . S . D a vid o v, J. H a t a m i

Yerevan State University
Department of Algebra and Geometry, Yerevan, Armenia

In t h e p r e s e n t p a p e r , we in t r o d u c e a lg e b r a s wit h fu z z y o p e r a t io n s . Th e p r o b le m o f d e ve l-
o p m e n t o f a lg e b r a s wit h fu z z y o p e r a t io n s is fo r m u la t e d in [1 ]. W e c o n s id e r fu z z y o p e r a t io n s
in s t e a d o f o r d in a r y fu n c t io n s in s t r u c t u r e o f a lg e b r a s .

1. I ntr oduction

Th e p r o b le m o f d e ve lo p m e n t o f a lg e b r a s wit h fu z z y o p e r a t io n s is fo r m u la t e d in ( [1 ], p .
1 3 6 ) . Fu z z y a p p r o a c h e s t o va r io u s u n ive r s a l a lg e b r a ic c o n c e p t s s t a r t e d wit h R o s e n fe ld `s
fu z z y g r o u p s [2 ]. S in c e t h e n , m a n y fu z z y a lg e b r a ic s t r u c t u r e s h a ve b e e n s t u d ie d . A ls o , s o m e
a u t h o r s p r o p o s e d a g e n e r a l a p p r o a c h t o t h e t h e o r y o f fu z z y a lg e b r a s . A n o t h e r fu z z y a p p r o a c h
t o u n ive r s a l a lg e b r a s wa s in it ia t e d b y B ·elo h la ve k a n d V yc h o d il [1 ,3 ], wh o s t u d ie d t h e s o -
c a lle d a lg e b r a s wit h fu z z y e qu a lit ie s a n d d e ve lo p e d a fu z z y e qu a t io n a l lo g ic . Th e s e s t r u c t u r e s
h a ve t wo p a r t s : t h e fu n c t io n a l p a r t , wh ic h is a n o r d in a r y a lg e b r a a n d t h e r e la t io n a l p a r t ,
wh ic h is t h e c a r r ie r s e t o f t h e a lg e b r a , e qu ip p e d wit h a fu z z y e qu a lit y wh ic h is c o m p a t ib le
wit h a ll o f t h e fu n d a m e n t a l o p e r a t io n s o f t h e c o r r e s p o n d in g a lg e b r a . In t h is p a p e r , we
in t r o d u c e a lg e b r a s wit h fu z z y o p e r a t io n s a n d wit h fu z z y e qu a lit ie s .

In fu z z y s e t t h e o r y t h e r e we r e d i®e r e n t a p p r o a c h e s t o t h e c o n c e p t o f a fu z z y fu n c t io n . In
a n u m b e r o f p a p e r s va r io u s kin d s o f fu z z y fu n c t io n s b a s e d o n fu z z y e qu iva le n c e r e la t io n s we r e
s t u d ie d . In p a r t ic u la r , s u c h a p p r o a c h wa s u s e d in d e ¯ n it io n s o f fu z z y fu n c t io n s a n d p a r t ia l
fu z z y fu n c t io n s , g ive n b y K la wo n n [4 ], s t r o n g fu z z y fu n c t io n s a n d p e r fe c t fu z z y fu n c t io n s ,
g ive n b y D e m ir c i [5 ]. In [6 ], a u t h o r s in ve s t ig a t e d kin d s o f a lg e b r a s wh ic h h a ve fu z z y s e t s
a s d o m a in s a n d o r d in a r y fu n c t io n s a s o p e r a t io n s . In [7 ], a u t h o r s d e ¯ n e d u n ifo r m fu z z y
r e la t io n a l m o r p h is m s b e t we e n t wo a lg e b r a s , a n d t h e n d e ve lo p e d fu z z y h o m o m o r p h is m s a n d
p r o ve d t h e o r e m s c o n c e r n in g t h e m . A le xa n d e r P . ·S o s t a k, in [8 ], s t u d ie d fu z z y c a t e g o r ie s wit h
fu z z y fu n c t io n s in t h e r o le o f m o r p h is m s

Fu z z y fu n c t io n s b a s e d o n fu z z y e qu iva le n c e r e la t io n s h a ve s h o wn t o b e ve r y u s e fu l in
m a n y a p p lic a t io n s in a p p r o xim a t e r e a s o n in g , fu z z y c o n t r o l, va g u e a lg e b r a a n d o t h e r ¯ e ld s .

4 9



5 0 Algebras with Fuzzy Operations

2. De¯nitions and Results

W e will u s e c o m p le t e r e s id u a t e d la t t ic e s L =< L; ^; _; ­; !; 0 ; 1 > a s t h e s t r u c t u r e s o f
t r u t h va lu e s .

De¯nition 1. L e t ¼M b e a fu z z y e qu a lit y o n M: A n ( n + 1 ) ¡ a r y fu z z y r e la t io n ½ o n
a s e t M is c a lle d a n n¡ a r y fu z z y o p e r a t io n w.r .t . ¼M a n d ¼M n if we h a ve t h e fo llo win g
c o n d it io n s

E xt e n s io n a lit y:

( p ¼M n p0 ) ­ ( y ¼M y0 ) ­ ½( p; y ) · ½( p0; y0 ) 8p; p0 2 M n; 8y; y0 2 M;

Fu n c t io n a lit y:

½( p; y ) ­ ½( p; y0 ) · y ¼M y0 8p 2 M n; 8y; y0 2 M;

Fu lly-d e fe n d :

_

y2M
½( p; y ) = 1 8p 2 M n;

wh e r e ( a1; :::; an ) ¼M
n

( b1; :::; bn ) =
Vn

i=1 ( ai ¼M bi ) : W e s a y t h a t ½ is a fu z z y o p e r a t io n o n
M wit h a r it y n:

De¯nition 2 [1 ]. A n a lg e b r a wit h L -e qu a lit y o r L ¡a lg e b r a o f t yp e <¼; F > is a t r ip le t
M = < M; ¼M ; F M > s u c h t h a t < M; F M > is a n a lg e b r a o f t yp e < F > a n d ¼M is a n
L¡ e qu a lit y o n M s u c h t h a t e a c h f M 2 F M is c o m p a t ib le wit h ¼M ; i.e .

( a1 ¼M b1 ) ­ ::: ­ ( an ¼M bn ) · f M ( a1; ::; an ) ¼M f M ( b1; :::; bn )
fo r e a c h n¡ a r y f 2 F a n d e ve r y a1; b1; :::; an; bn 2 M:

De¯nition 3. A n a lg e b r a wit h fu z z y o p e r a t io n s o f t yp e <¼; F > is a t r ip le t M =<
M; ¼M ; FM > s u c h t h a t

( i) ¼M is a fu z z y e qu a lit y o n t h e s e t M;
( ii) FM is t h e s e t o f fu z z y o p e r a t io n s o n t h e s e t M:
To s im p ly, we c a ll F-a lg e b r a s in s t e a d o f t h e a lg e b r a wit h fu z z y o p e r a t io n s .
T heor em 1. L e t L b e a H e yt in g a lg e b r a ( ^ = ­ ) a n d M =< M; ¼M ; F M > b e a n

L ¡ a lg e b r a o f t yp e <¼; F > : L e t f M : M n £M ! L wit h f M ( p; y ) = f M ( p) ¼M y fo r e ve r y
n¡ a r y f M 2 F M a n d fo r a ll p 2 M n; y 2 M: Th e n M =< M; ¼M ; F M > is a n F ¡a lg e b r a
o f t yp e <¼; F > :

De¯nition 4. L e t M =< M; ¼M ; FM > b e a n F ¡a lg e b r a o f t yp e <¼; F > : A n
L ¡ r e la t io n ( b in a r y fu z z y r e la t io n ) µ o n M is c a lle d a c o n g r u e n c e o n M if

( i) µ is a fu z z y e qu iva le n c e o n M ;
( ii) µ is c o m p a t ib le wit h ¼M ; i.e . ( a ¼M b ) ­ ( a0 ¼M b0 ) ­ µ ( a; a0 ) · µ ( b; b0 ) fo r e ve r y

a; a0; b; b0 2 M ;
( iii) ^ni=1µ ( ai; bi ) ­ µ ( y; y0 ) ­ f M ( a1; :::; an; y ) · f M ( b1; :::; bn; y0 ) fo r e ve r y n¡a r y fu z z y

o p r a t io n f M 2 FM a n d a1; :::; an; y; y0 2 M:
Th e o r d in a r y s e t o f a ll c o n g r u e n c e s o n F ¡a lg e b r a M is d e n o t e d b y Con( M) :
De¯nition 5. L e t µ b e a c o n g r u e n c e o n a n F ¡ a lg e b r a M =< M; ¼M ; FM > o f t yp e

<¼; F > : A fa c t o r a lg e b r a o f M b y µ is a n F ¡a lg e b r a M=µ =< M=µ; ¼M=µ; FM=µ > o f
t yp e <¼; F > s u c h t h a t



S. Davidov, J. Hatami 5 1

( i) [a]µ ¼M=µ [b]µ = µ ( a; b) fo r e a c h [a]µ; [b]µ 2 M=µ;
( ii) f M=µ ( [a1]µ; :::; [an]µ; [y]µ ) = f

M ( a1; :::; an; y )
fo r e ve r y n¡ a r y o p e r a t io n f M=µ 2 FM=µ a n d fo r a r b it r a r y a1; :::; an; y 2 M, wh e r e M=µ =

f[a]µj a 2 Mg a n d [a]µ = fa0j µ ( a; a0 ) = 1 g:
De¯nition 6 [1 ]. L e t M a n d N b e L ¡ a lg e b r a s o f t yp e <¼; F > : A m a p p in g h : M ! N

is c a lle d a m o r p h is m o f M t o N if
( i) h( f M ( a1; :::; an ) ) = f

N ( h( a1 ) ; :::; h( an ) ) fo r e ve r y n¡a r y f 2 F a n d a r b it r a r y
a1; :::; an 2 M;

( ii) a ¼M b · h( a ) ¼N h( b ) fo r e ve r y a; b 2 M:
If a ¼M b = h( a ) ¼M h ( b ) fo r e ve r y a; b 2 M; t h e n h is c a lle d a n e m b e d d in g o f M t o N :
De¯nition 7. L e t M =< M; ¼M ; FM > a n d N =< N; ¼N ; FN > b e t wo F ¡ a lg e b r a s

o f t yp e <¼; F > : A m a p p in g h : M ! N is c a lle d a m o r p h is m o f M t o N if
( i) a ¼M b · h( a) ¼N h( b ) fo r a ll a; b 2 M;
( ii) f M ( a1; :::; an; y ) = f

N ( h ( a1 ) ; :::; h( an ) ; h( y ) ) fo r e ve r y n¡a r y o p e r a t io n f M 2 FM a n d
a r b it r a r y a1; :::; an; y 2 M:

A m o n o m o r p h is m is a n in je c t ive m o r p h is m , a n e m b e d d in g is a m o r p h is m s u c h t h a t , fo r
e ve r y a; b 2 M;

a ¼M b = h( a ) ¼N h( b ) ;
fo r a ll f M 2 FM wit h a r it y n a n d fo r a ll a1; :::; an 2 M:

A n e p im o r p h is m wh ic h is a n e m b e d d in g is c a lle d a n is o m o r p h is m .
T heor em 2. L e t L b e a H e yt in g a lg e b r a . L e t M a n d N b e t wo L ¡ a lg e b r a s o f t yp e

<¼; F > a n d a m a p p in g h : M ! N b e a n e m b e d d in g o f M t o N : Th e n h is a n e m b e d d in g
o f M t o N

De¯nition 8. Fo r e ve r y F ¡a lg e b r a s M a n d µ 2 Con( M) a m a p p in g hµ : M ! M=µ;
wh e r e hµ ( a) = [a]µ fo r a ll a 2 M; is c a lle d a n a t u r a l m a p p in g .

Lemma 1. A n a t u r a l m a p p in g hµ fr o m F ¡ a lg e b r a s M t o a fa c t o r F ¡a lg e b r a M=µ is
a n e p im o r p h is m .

T heor em 3. ( ¯ r s t is o m o r p h is m t h e o r e m ) . L e t h : M ! N b e a n e p im o r p h is m o f
F ¡ a lg e b r a s . Th e n t h e r e is a n is o m o r p h is m g : M=µh ! N s u c h t h a t hµh ± g = h, wh e r e
µh = f( a; b ) jh( a) ¼N h( b ) g:

De¯nition 9. L e t M b e a n F ¡ a lg e b r a a n d Á; µ 2 Con( M) ; µ µ Á: Th e n we le t Á=µ
d e n o t e a n L ¡ r e la t io n o n M=µ d e ¯ n e d b y ( Á=µ ) ( [a]µ; [b]µ ) = Á( a; b ) fo r a ll a; b 2 M:

T heor em 4. L e t M b e a n F ¡a lg e b r a a n d Á; µ 2 Con( M) ; µ µ Á: Th e n Á=µ 2
Con( M=µ ) :

T heor em 5. ( s e c o n d is o m o r p h is m t h e o r e m ) . S u p p o s e M is a n F ¡ a lg e b r a a n d Á; µ 2
Con( M) ; µ µ Á: Th e n t h e m a p p in g

h : ( M=µ ) = ( Á=µ ) ! M=Á

d e ¯ n e d b y h( [[a]µ]Á=µ ) = [a]Á is a n is o m o r p h is m .

R e fe r e n c e s

[1 ] B ·elo h la ve k R ., V yc h o d il V ., Fu z z y e qu a t io n a l lo g ic , S p r in g e r , 2 0 0 5 .

[2 ] R o s e n fe ld A ., Fu z z y g r o u p s , J. Ma t h . A n a l. A p p l. 3 5 ( 3 ) ( 1 9 7 1 ) , 5 1 2 -5 1 7 .



5 2 Algebras with Fuzzy Operations

[3 ] B ·elo h la ve k R ., V yc h o d il V ., A lg e b r a s wit h fu z z y e qu a lit ie s , Fu z z y S e t s a n d S ys t e m s ,
1 5 7 ( 2 0 0 6 ) , 1 6 1 -2 0 1 .

[4 ] K la wo n n F., Fu z z y p o in t s , in : V . N o va k, I. P e r ¯ lie va ( E d s ) , D is c o ve r in g W o r ld wit h
Fu z z y L o g ic , P h ys ic a , H e id e lb e r g , ( 2 0 0 0 ) , 4 3 1 -4 5 3 .

[5 ] D e m ir c i M., Fu z z y fu n c t io n s a n d t h e ir fu n d a m e n t a l p r o p e r t ie s , Fu z z y S e t s a n d S ys t e m s ,
1 0 6 ( 1 9 9 9 ) , 2 3 6 -2 4 6 .

[6 ] B o ·sn ja k I., Ma d a r a s z R ., V o jo d ic G., A lg e b r a s o f fu z z y s e t s , Fu z z y S e t s a n d S ys t e m s ,
1 6 0 ( 2 0 0 9 ) , 2 9 7 9 -2 9 8 8 .

[7 ] Ig n ja t o vic J., Cir ic M., B o g d a n o vic S ., Fu z z y h o m o m o r p h is m s o f a lg e b r a s , Fu z z y S e t s
a n d S ys t e m s , 1 6 0 ( 2 0 0 9 ) , 2 3 4 5 -2 3 6 5 .

[8 ] ·S o s t a k A .P ., Fu z z y fu n c t io n s a n d e xt e n s io n o f t h e c a t e g o r y L -To p o f CH A N G-GOGU E N
L -t o p o lo g ic a l s p a c e s , 2 7 1 -2 9 4 .