D:\User\sbornik_38_pdf\22.DVI


Mathematical Problems of Computer Science 38, 53{55, 2012.

On M edial-like Functional E quations

A m ir E h s a n i

Department of Mathematics
Mahshahr Branch, Islamic Azad University

Mahshahr, Iran.
Mahshahr Branch, Islamic Azad University, Mahshahr, Iran.

a.ehsani@mahshahriau.ac.ir

L e t A b e a n o n e m p t y s e t , n a n d m b e p o s it ive in t e g e r s a n d f : An ! Am b e a n a r b it r a r y
fu n c t io n . Th e n ( A; f ) is c a lle d [n; m]-g r o u p o id . Th e n-a r y o p e r a t io n s , f1; : : : ; fm, a r e d e ¯ n e d
b y t h e fo llo win g :

f ( x1; : : : ; xn ) = ( y1; : : : ; ym ) , yi = fi ( x1; : : : ; xn ) ;

fo r e ve r y 1 · i · m, a r e c a lle d t h e c o m p o n e n t o p e r a t io n s o f f a n d t h e y a r e d e n o t e d b y
f = ( f1; : : : ; fm ) [1 1 , 1 2 , 1 3 ]. Th e [n; m]-g r o u p o id is p r o p e r i® n; m; jQj ¸ 2 .

Th e [n; m]-g r o u p o id ( A; f ) is c a lle d [n; m]-qu a s ig r o u p ( o r m u lt iqu a s ig r o u p [2 , 3 , 1 4 ]) i®
fo r e ve r y in je c t io n , Á : Nn ! Nn+m, wh e r e Nn = f1 ; : : : ; ng, a n d e ve r y ( a1; : : : ; an ) 2 Qn
t h e r e e xis t s a u n iqu e ( b1; : : : ; bn+m ) 2 Qn+m s u c h t h a t :

f ( b1; : : : ; bn ) = ( bn+1; : : : ; bn+m ) a n d bÁ(i) = ai;

fo r i = 1 ; : : : ; n.
It is c le a r t h a t Q( f ) is a n [n; 1 ]-qu a s ig r o u p i® Q( f ) is a n n-qu a s ig r o u p [1 ]. Q( f ) is a [1 ; m]-

qu a s ig r o u p i® t h e r e e xis t p e r m u t a t io n s , f1; : : : ; fm, o f Q s u c h t h a t f ( x ) = ( f1 ( x ) ; : : : ; fm ( x) ) .
It is a ls o c le a r t h a t a ll c o m p o n e n t s o f a m u lt iqu a s ig r o u p a r e b in a r y qu a s ig r o u p o p e r a t io n s .

If t h e c o m p o n e n t o p e r a t io n s o f t h e [n; m]-qu a s ig r o u p a r e b in a r y o p e r a t io n s , i.e . n = 2 ,
t h e n we s a y t h a t t h e [n; m]-qu a s ig r o u p is a b in a r y m u lt iqu a s ig r o u p .

L e t u s c o n s id e r t h e fo llo win g h yp e r id e n t it ie s [7 , 8 , 9 ]:

g ( f ( x; y ) ; f ( u; v ) ) = f ( g ( x; u ) ; g ( y; v ) ) ; ( Me d ia lit y)

g ( f ( x; y ) ; f ( u; v ) ) = f ( g ( v; y ) ; g ( u; x) ) ; ( P a r a m e d ia lit y)

g ( f ( x; y ) ; f ( u; v ) ) = g ( f ( x; u ) ; f ( y; v ) ) ; ( Co -m e d ia lit y)

g ( f ( x; y ) ; f ( u; v ) ) = g ( f ( v; y ) ; f ( u; x) ) ; ( Co -p a r a m e d ia lit y)

f ( x; x ) = x: ( Id e m p o t e n c y)

Th e b in a r y a lg e b r a , ( A; F ) , is c a lle d :

² m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .1 ) ,

² p a r a m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .2 ) ,

5 3



5 4 On Medial-like Functional Equations

² c o -m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .3 ) ,

² c o -p a r a m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .4 ) ,

² id e m p o t e n t , if it s a t is ¯ e s t h e id e n t it y ( 1 .5 ) ,

fo r e ve r y f; g 2 F . Th e b in a r y a lg e b r a , ( A; F ) , is c a lle d m o d e , if it is m e d ia l a n d id e m p o t e n t .

De¯nition 1 The binary multiquasigroup ( A; f ) with f = ( f1; : : : ; fm ) is called:

² medial, if the binary algebra, ( A; f1; : : : ; fm ) , is medial,

² paramedial, if the binary algebra, ( A; f1; : : : ; fm ) , is paramedial,

² co-medial, if the binary algebra, ( A; f1; : : : ; fm ) , is co-medial,

² co-paramedial, if the binary algebra, ( A; f1; : : : ; fm ) , is co-paramedial,

² idempotent, if the binary algebra, ( A; f1; : : : ; fm ) , is idempotent,

² mode, if the binary algebra, ( A; f1; : : : ; fm ) , is a mode.

Th e n e xt c h a r a c t e r iz a t io n o f b in a r y m e d ia l m u lt iqu a s ig r o u p s fo llo ws fr o m [6 , 1 0 ].

T heor em 1 L et ( Q; f ) be a binary multiquasigroup, where f = ( f1; : : : ; fm ) . If ( Q; f ) is a
binary medial multiquasigroup, then there exists an abelian group, ( Q; +) , such that:

fi ( x; y ) = ®ix + ¯iy + ci;

where ®i; ¯i are automorphisms of the group ( Q; +) , and ci 2 Q is a ¯xed element and:
®i¯j = ¯j®i; ®i®j = ®j®i; ¯i¯j = ¯j¯i, for i; j = 1 ; : : : ; m. The group, ( Q; +) , is unique up
to isomorphisms. M oreover, if ( Q; f ) is a mode, then

fi ( x; y ) = ®ix + ¯iy;

where ®i; ¯i are automorphisms of both the group, ( Q; +) , and of the algebra, ( Q; f1; : : : ; fm ) .

In t h is p a p e r we c h a r a c t e r iz e t h e b in a r y p a r a m e d ia l, c o -m e d ia l a n d c o -p a r a m e d ia l m u l-
t iqu a s ig r o u p s ( c f. [4 , 5 ]) .

R e fe r e n c e s

[1 ] B e lo u s o v, V . D . ( 1 9 7 2 ) . n-a r y qu a s ig r o u p s . In : Shtiinca, K is h in e v.

[2 ] ·C u p o n a , G., U ·sa n , J., S t o ja ko vi¶c, Z. ( 1 9 8 0 ) . Mu lt iqu a s ig r o u p s a n d s o m e r e la t e d s t r u c -
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[3 ] ·C u p o n a , G., S t o ja ko vi¶c, Z., U ·sa n , J. ( 1 9 8 1 ) . On ¯ n it e m u lt iqu a s ig r o u p s . P ubl. Inst.
M ath. V o l. 2 9 , N o . 4 3 , 5 3 -5 9 .

[4 ] E h s a n i, A ., Mo vs is ya n , Y u . M. ( 2 0 1 2 ) . L inear representation of medial-like algebras.
Co m m u n ic a t io n s in A lg e b r a .



A. Ehsani 5 5

[5 ] E h s a n i, A ., Mo vs is ya n , Y u . M. ( 2 0 1 2 ) . B inary M ultiquasigroups with M edial-L ike E qua-
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[7 ] Mo vs is ya n , Y u . M. ( 1 9 9 0 ) . H yp e r id e n t it ie s a n d h yp e r va r ie t ie s in a lg e b r a s . In : Yerevan
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[9 ] Mo vs is ya n , Y u . M. ( 1 9 8 6 ) . In t r o d u c t io n t o t h e t h e o r y o f a lg e b r a s wit h h yp e r id e n t it ie s .
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[1 1 ] P o lo n ijo , M. ( 1 9 8 1 ) . S t r u c t u r e t h e o r e m fo r Cn+1-s ys t e m s . Glasnik M at. V o l. 1 6 , N o . 3 8 ,
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[1 2 ] P o lo n ijo , M. ( 1 9 8 2 ) . On Cn+1-s ys t e m s a n d [n; m]-g r o u p o id s . Algebraic Conference,
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[1 3 ] S t o ja ko vi¶c, Z. ( 1 9 8 2 ) . On b is ym e t r ic [n; m]-g r o u p o id s . R eview of research F aculty of
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[1 4 ] S t o ja ko vi¶c, Z., P a u n i¶c, D j. ( 1 9 8 2 ) . Id e n t it ie s o n m u lt iqu a s ig r o u p s . P roc. Symp. On n-ary
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