D:\User\sbornik_38_pdf\33.DVI


Mathematical Problems of Computer Science 38, 77{79, 2012.

On the Algebr as with H yper identities of the Var iety

of de M or gan Algebr as

Y u . M. Mo vs is ya n , V . A . A s la n ya n

Yerevan State University,
Alex Manoogian 1, Yerevan 0025, Armenia

E-mail: yurimovsisyan@yahoo.com, vahagn.aslanyan@gmail.com

1 In t r o d u c t io n

In p a p e r [1 ] t h e a lg e b r a s wit h h yp e r id e n t it ie s o f t h e va r ie t y o f B o o le a n a lg e b r a s a r e c h a r a c -
t e r iz e d . In t h is p a p e r t h e a lg e b r a s wit h h yp e r id e n t it ie s o f t h e va r ie t y o f D e Mo r g a n a lg e b r a s
a r e c h a r a c t e r iz e d . Fo r t h e s e a lg e b r a s wit h t wo b in a r y o p e r a t io n s we p r o ve a s t r u c t u r e r e s u lt .
A s a c o n s e qu e n c e , we o b t a in t h e n e w ¯ n it e b a s e o f t h e h yp e r id e n t it ie s o f t h e va r ie t y o f D e
Mo r g a n a lg e b r a s , h a vin g fu n c t io n a l a n d o b je c t ive r a n ks n o t e xc e e d in g t h r e e .

A n a lg e b r a Q( +; ¢;0 ) wit h t wo b in a r y a n d o n e u n a r y o p e r a t io n s is c a lle d a D e Mo r g a n
a lg e b r a if Q ( +; ¢ ) is a d is t r ib u t ive la t t ic e a n d Q ( +; ¢;0 ) s a t is ¯ e s t h e fo llo win g id e n t it ie s :

( x + y ) 0 = x0 ¢ y0;

x00 = x;

wh e r e x00 = ( x0 ) 0. Th e s t a n d a r d fu z z y a lg e b r a F = ( ( 0 ; 1 ) ; max ( x; y ) ; min( x; y ) ; 1 ¡ x ) is a n
e xa m p le o f a D e Mo r g a n a lg e b r a .

D e Mo r g a n a lg e b r a s we r e c o n s id e r e d b y J.A .K a lm a n [2 ]( a s i-la t t ic e s ) , G.C.Mo is il [3 ],
H .R a s io wa a n d A .B ia lyn ic ki-B ir u la [4 ], Y u .M.Mo vs is ya n [5 ], J. B e r m a n a n d W . B lo k [6 ]
a n d o t h e r s . Th e y a ls o r e la t e d t o c o n s t r u c t ive lo g ic wit h s t r o n g n e g a t io n ( A .A .Ma r ko v [7 ],
D .N e ls o n [8 ], N .N .V o r o b e v [9 ], I.D .Za s la vs ky [1 0 ]) . E xc e p t in m a t h e m a t ic a l lo g ic a n d a lg e -
b r a , D e Mo r g a n a lg e b r a s ( a n d D e Mo r g a n b is e m ila t t ic e s ) h a ve a p p lic a t io n s in m u lt i-va lu e d
s im u la t io n s o f d ig it a l c ir c u it s t o o ( [1 1 , 1 2 ]) .

Th e h yp e r id e n t it ie s o f t h e va r ie t y o f D e Mo r g a n a lg e b r a s a r e c h a r a c t e r iz e d in [1 3 ].

De¯nition 1.1 A T -algebra A = ( Q; §) , where T = f1 ; 2 g, is called D e M organ quasilattice
if it satis¯es all hyperidentities of the variety of D e M organ algebras.

Fo r e xa m p le , t h e s u p e r p r o d u c t ( [1 4 , 1 5 , 1 6 , 1 7 ]) o f t h e t wo D e Mo r g a n a lg e b r a s ( D e
Mo r g a n qu a s ila t t ic e s ) is a D e Mo r g a n qu a s ila t t ic e .

7 7



7 8 On the Algebras with Hyperidentities of the Variety of de Morgan Algebras

2 Ma in r e s u lt

B e lo w we d e ¯ n e t h e c o n c e p t o f D e Mo r g a n s u m a n a lo g o u s t o B o o le a n s u m in t r o d u c e d in [1 ].

De¯nition 2.1 L et A = ( Q; ­ [ fF g) be an algebra with a unary operation F . L et
( Qi;­ ) ; i 2 I be subalgebras of the algebra A, and Ai = ( Qi; ­ [ fFig) be algebras with
a unary operation Fi. The algebra A is called D e M organ sum of algebras Ai, if the following
conditions hold true:
1 ) Qi \ Qj = Â for all i; j 2 I; i 6= j;
2 ) Q =

S
i2I Qi;

3 ) Two binary operations +; ¢ and a unary operation ¹ can be de¯ned on I such that I ( +; ¢; ¹ )
is a D e M organ algebra;
4 ) If i; j 2 I and i · j (here " · " is the order of the lattice I ( +; ¢ ) ), then there exists an
isomorphism

( 'i;j; ~" ) : Ai ! Aj ;
where ~" ( Fi ) = Fj; ~" ( A ) = A for any A 2 ­. M oreover, 'i;i is the identical mapping of the
set Qi, and for all i · j · k we have 'i;j ¢ 'j;k = 'i;k;
5 ) F or every i 2 I there exists an isomorphism

( hi;i; ~" ) : Ai ! Ai;

such that h¡1
i;i

= hi;i and hi;i ¢ 'i;k = 'i;k for all k ¸ i + i; k 2 I;
6 ) F or any n-ary operation A 2 ­ ( n ¸ 2 ) and for any x1; : : : ; xn 2 Q we have:

A ( x1; : : : ; xn ) = A ( 'i1;i0 ( x1 ) ; : : : ; 'in;i0 ( xn ) ) ;

where xj 2 Qij , ij 2 I; j = 1 ; n, i0 = i1 + : : : + in;
7 ) F or any x 2 Q we have:

F ( x ) = hi;i ( Fi ( x ) ) ;

where x 2 Qi.

T heor em 2.1 An algebra A = ( Q; f+; ¢;¹ g) with two binary operations +; ¢ and one unary
operation ¹ is a D e M organ quasilattice i® it is a D e M organ algebra or D e M organ sum of
D e M organ algebras.

Cor ollar y 2.1 The variety of D e M organ algebras has a ¯nite base of hyperidentities having
functional and objective ranks not exceeding three.

R e fe r e n c e s

[1 ] Y u .M. Mo vs is ya n , Algebras with hyperidentities of the variety of B oolean algebras.
Iz ve s t iya R o s s iys ko y A ka d e m ii N a u k: S e r iya Ma t e m a t ic h e s ka ya 6 0 , 1 2 7 -1 6 8 , 1 9 9 6 . E n -
g lis h t r a n s la t io n in R u s s ia n A c a d e m y o f S c ie n c e Iz ve s t iya Ma t e m a t is c h e s ka ya , 6 0 , 1 2 1 9 -
1 2 6 0 , 1 9 9 6 .

[2 ] J. A . K a lm a n , L a t t ic e s wit h in vo lu t io n , Trans. Amer. M ath. Soc. 8 7 , ( 1 9 5 8 ) , 4 8 5 -4 9 1 .

[3 ] G.C.Mo is il, R e c h e r c h e s s u r l'a lg e b r e d e la lo g iqu e , Annales scienti¯ques de l'universite
de J assy, 2 2 , ( 1 9 3 5 ) , 1 -1 1 7 .



Yu. Movsisyan, V. Aslanyan 7 9

[4 ] A .B ia lyn ic ki-B ir u la , H .R a s io wa , On t h e r e p r e s e n t a t io n o f qu a s i-B o o le a n a lg e b r a s , B ull.
Acad. P olon. Sci., Ser. M ath. Astronom. P hys., 5 , ( 1 9 5 7 ) , 2 5 9 -2 6 1 .

[5 ] Y u .M.Mo vs is ya n , B in a r y r e p r e s e n t a t io n s o f a lg e b r a s wit h a t m o s t t wo b in a r y o p e r a -
t io n s . A Ca yle y t h e o r e m fo r d is t r ib u t ive la t t ic e s , International J ournal of Algebra and
Computation, V o l.1 9 , 1 ( 2 0 0 9 ) , 9 7 -1 0 6 .

[6 ] J. B e r m a n , W . B lo k, S t ip u la t io n s , m u lt i-va lu e d lo g ic a n d D e Mo r g a n a lg e b r a s , M ulti-
valued L ogic 7 ( 5 -6 ) ( 2 0 0 1 ) , 3 9 1 -4 1 6 .

[7 ] A .A .Ma r ko v, Co n s t r u c t ive L o g ic ( in R u s s ia n ) ,Uspekhi M at. Nauk, 5 ( 1 9 5 0 ) ,1 8 7 -1 8 8 .

[8 ] D .N e ls o n , Co n s t r u c t ib le fa ls it y, J . Symbolic L ogic, 1 4 ( 1 9 5 9 ) , 1 6 -2 6 .

[9 ] N .N .V o r o b e v, A c o n s t r u c t ive p r o p o s it io n a l c a lc u lu s wit h s t r o n g n e g a t io n ( in R u s s ia n ) ,
D okl. Akad. Nauk SSR , 8 5 ( 1 9 5 2 ) , 4 6 5 -4 6 8 .

[1 0 ] I.D .Za s la vs ky, Symmetric Constructive L ogic ( in R u s s ia n ) , P u b lis h in g H o u s e o f
A c a d e m y o f S c ie n c e s o f A r m e n ia n S S R ( 1 9 7 8 ) .

[1 1 ] J.A . B r z o z o ws ki, D e Mo r g a n b is e m ila t t ic e s , P roceedings of the 30th IE E E International
Symposium on M ultiple-Valued L ogic ( IS MV L 2 0 0 0 ) , Ma y 2 3 -2 5 , ( 2 0 0 0 ) , p .1 7 3 .

[1 2 ] J.A . B r z o z o ws ki, P a r t ia lly o r d e r e d s t r u c t u r e s fo r h a z a r d d e t e c t io n , Special Session: The
M any L ives of L attice Theory, J oint M athematics M eetings, S a n D ie g o , CA , Ja n u a r y
6 -9 , ( 2 0 0 2 ) .

[1 3 ] Y u .M.Mo vs is ya n , V .A .A s la n ya n , Hyperidentities of D e M organ algebras, L o g ic Jo u r n a l
o f t h e IGP L .( d o i:1 0 .1 0 9 3 / jig p a l/ jz r 0 5 3 )

[1 4 ] Y u .M. Mo vs is ya n , Introduction to the theory of algebras with hyperidentities ( in R u s -
s ia n ) , Y e r e va n S t a t e U n ive r s it y P r e s s , Y e r e va n , 1 9 8 6 .

[1 5 ] Y u .M. Mo vs is ya n , H yp e r id e n t it ie s in a lg e b r a s a n d va r ie t ie s , Uspekhi M atematicheskikh
Nauk, V o l.5 3 , N o . 1 ( 3 1 9 ) , ( 1 9 9 8 ) , 6 1 { 1 1 4 . E n g lis h t r a n s la t io n in R u s s ia n Ma t h e m a t ic a l
S u r ve ys 5 3 , 1 , ( 1 9 9 8 ) , 5 7 { 1 0 8 .

[1 6 ] Y u .M.Mo vs is ya n , B ila t t ic e s a n d h yp e r id e n t it ie s , P roceedings of the Steklov Institute of
M athematics, V o l. 2 7 4 , ( 2 0 1 1 ) , p p . 1 7 4 -1 9 2 .

[1 7 ] Y u .M.Mo vs is ya n , A .B .R o m a n o ws ka , J.D .H S m it h , S u p e r p r o d u c t s , h yp e r id e n t it ie s , a n d
a lg e b r a ic s t r u c t u r e s o f lo g ic p r o g r a m m in g , Comb. M ath. And Comb. Comp., 2 0 0 6 , v.5 8 ,
p .1 0 1 -1 1 1 .