S A T U R A T I O N O F B O O L E A N A L G E B R A S

A . I . O m a r o v UDC 519 .49

In the p r e s e n t note we g ive a c h a r a c t e r i s t i c of the s a t u r a t i o n of B o o l e a n a l g e b r a s in t e r m s of the e l e m e n t a r y c l a s s i f i c a t i o n of Yu. L . E r s h o v [1].

Le t us i n t r o d u c e n e c e s s a r y d e f i n i t i o n s . L e t B be a B o o l e a n a l g e b r a and r be an e l e m e n t a r y type of B o o l e a n a l g e b r a s . We s h a l l denote by t r ( a ) the e l e m e n t a r y type of an e l e m e n t a [2] and by Mr the s e t of a l l t hose e l e m e n t s of the B o o l e a n a l g e b r a B which have the type T. F u r t h e r l e t ~- be an e l e m e n t a r y type of the f o r m ( a t , 0, n 2 ) , ( n l , ~ , n 2) o r (~o, 0, 0 ) . F o r t ~ ( ~ , 0, 0> we d e n o t e b y Mr the s e t of the e l e - m e n t s of the B o o l e a n a l g e b r a B wi th the e l e m e n t a r y t y p e s of the f o r m ( t r I ( r ) , n, 0 }, w h e r e n is a p o s i t i v e i n t e g e r , and M ( ~ , 0, o) : B .

L e t k be a t r a n s f i n i t e c a r d i n a l n u m b e r . We s h a l l c a l l a s u b s e t X of the Boo l e a n a l g e b r a B a f i n i t e ly r - i n t e r s e c t i n g s u b s e t if X ~-Mr and t r ( x I N . . . N Xn) ~: r f o r e v e r y n and a r b i t r a r y e l e m e n t s x~ . . . . . Xn~ X,

A Boo lean a l g e b r a B is c a l l e d k - c o m p a c t i f f o r an a r b i t r a r y c a r d i n a l n u m b e r 3' < k and e v e r y type Z of the f o r m u n d e r c o n s i d e r a t i o n and an a r b i t r a r y f i n i t e l y r - i n t e r s e c t i n g s u b s e t of e a r d i n a l i t y 3' t h e r e e x i s t s an e l e m e n t b 6 M r such tha t b = x fo r e v e r y x~ X.

A s e t X of e l e m e n t s of a B o o l e a n a l g e b r a B is c a l l e d k - s e p a r a b l e if f o r a r b i t r a r y / 3 < k and a r b i t r a r y s u b s e t s {a 7 IY </3 } -= X and {by 17 </3} ~ B t h e r e e x i s t s an u p p e r bound a of the e l e m e n t s {a 7 17 </3} in the s e t X such tha t i f b 7 c: a 6 : ~ f o r e v e r y 5 </3, then b 7 n a = e . A B o o l e a n a l g e b r a B i s c a l l e d k - s e p a r a b l e i f f o r e v e r y e l e m e n t a r y type r the s e t s M T and M r U Mr a r e k - s e p a r a b l e .

Le t us c h o o s e a s y s t e m of e l e m e n t s {ac~ I c~ </3 } of a B o o l e a n a l g e b r a B and l e t A be the s u b a l g e b r a g e n e r a t e d by th i s s y s t e m of e l e m e n t s . I t i s e a s y to p r o v e the fo l lowing l e m m a .

L E M M A . Two e l e m e n t s x and y of a B o o l e a n a l g e b r a B a r e i n d i s t i n g u i s h a b l e r e l a t i v e to a s y s t e m of e l e m e n t s { a~ I a </3 ~ if and only i f f o r e v e r y a in A

tr(xNa)-:-tr(yfia), t r(qxfla) = t r (qyNa) .

T H E O R E M . A B o o l e a n a l g e b r a B is k - s a t u r a t e d i f and only i f i t i s k - c o m p a c t and k - s e p a r a b l e .

P r o o f . S ince the k - c o m p a c t n e s s and the k - s e p a r a b i l i t y can be w r i t t e n down by the f o r m u l a s of a g e n e r a l - p u r p o s e f e e d e r , the n e c e s s i t y of the cond i t ion i s o b v i o u s . L e t us p r o v e the s u f f i c i e n c y of the con - d i t i on . L e t Z be a f i n i t e l y r e a l i z e d s e t of f o r m u l a s in B of c a r d i n a l i t y / 3 ; then i t is r e a l i z e d in an e l e m e n - t a r y e x t e n s i o n B' of the a l g e b r a B. Le t an e l e m e n t x ~ B ' s a t i s f y a l l the f o r m u l a s of Z and l e t A be the s u b - a l g e b r a g e n e r a t e d in B by the e l e m e n t s of the s e t {ac~ la </3 } whose s y m b o l s o c c u r in the f o r m u l a s of ~ . By v i r t u e of L e m m a 1 i t i s s u f f i c i e n t to f ind a y in B such tha t t r ( x Na) = t r ( y fla) and t r (-1 x fia) - t r (q y

a) f o r e v e r y a in A .

L e t a be an e l e m e n t of B ' . If t r ( a ) has one of the f o r m s <n, 1, 0 ) , (n , 0, 1 ) , (n , ~o, 0 ) , (oo, 0, 0 ) , then f o r e v e r y e l e m e n t b of A e i t h e r t r (a fib) = t r ( a ) o r (a n q b ) = t r ( a ) . T h e r e f o r e t h e r e e x i s t s an u l t r a f i l t e r P a o v e r A such tha t c ~ P a ~ t r ( a tic) = t r ( a ) . Le t us put F a = { P a } - If t r ( a ) has a d i f f e r e n t f o r m <n 1, n 2, n 3} o r (n , ~ , 1 ) , then a can be d e c o m p o s e d into a f in i te n u m b e r of p a r t s Ca t . . . . . c a m such tha t Ca t . . . . Ca m have the f o r m s a l r e a d y c o n s i d e r e d . Then l e t F a - { P e a l , " " , PCa m ~ "

Le t F A be the s e t of u l t r a f i l t e r s o v e r A de f ined by the i m b e d d i n g of the a l g e b r a A in the c o m p l e t e Boo l ean a l g e b r a 2 I, w h e r e III = IAI , i . e . FA = { p i , i E I } , a 6 P i ~=~iEa, a 6 A . Le t F : F A UaUexF a . It

T r a n s l a t e d f r o m S i b i r s k i i M a t e m a t i c h e s k i i Z h u r n a l , Vol . 15, No. 6, pp . 1414-1415, N o v e m b e r -

D e c e m b e r , 1974. O r i g i n a l a r t i c l e s u b m i t t e d O c t o b e r 17, 1973.

0 1975 Ph'mon Publishing Cr 227 West 17lh Street. New York, N. Y. lOOI 1. N~ p~mt ,~] this puhliratio~l may he reproduced. ste.'ed in u re t r ieva l sl,slcD1, o r trt l l lSmillr in tlll.F f~)lTH OF ])V alll ' tile(IllS, e l e c t r o n i c , tII('ch(IIIicH]. I~ho&wOl~)'iug. micro fihnhtg,

recordittg or otherwisel witlu~ttt wJittett permission o f the imhlisher. :1 ct)py o f this arti~'h' ix at,ailahh' from the imhlishcr fi ~r $15.00.

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~fL is c l ea r that [ r I < k. F o r bE B' let ra, b = ~=~U r e a l nb , r b : r A u{b} U . ~ r a , b.

By virtue of the k -compac tness of B all the u l t raf i l te rs of r a re rea l ized in B. Let ( c a I a < ~ ~ be all the elements real iz ing the u l t ra f i l te rs of r . Every element c a can be represen ted in the form of the

join of two disjoint e lements caX and ca ~ x real iz ing the corresponding u l t raf i l te rs r x and I" ~ x By virtue

of the k-separab i l i ty of B there exists an e lement dE B such that ca x _=d, c a ~x N d = ~ for every a < ft.

Let us show that t r ( x Aa) = t r ( d Na). F o r the e lement a there exist elements ca1 . . . . . Cam such that t r (a) = t r ( c a l U �9 �9 �9 U Cam).

Then by virtue of the fact that d _ c a . x U �9 �9 �9 U ca... x we have t r (a fix) <- t r ( a Ndt. Since every e le- ment b a is decomposed into elements real iz ing u l t raf i l te rs of I? A, it follows that t r ( a fi d) -< t r ( a N x), i . e . , t r (x O at = t r (d N a). Let us now show that t r ( u x Nat = t r ( ~ d Oa). It is c lea r that Ud is an upper bound of the elements {c a ix i a < fl} disjoint f rom ( c a x [c~ < f~}. Therefore an analogous reasoning proves the validity of the equation tr(-1 x Nat = t r ( -Td Nat for every aEA. The theorem is proved.

The author thanks the part icipants of the s emina r "Model Theory" oZ the Mathematics Institute of the Siberian Section of the Academy of Sciences of the USSR for pointing out omiss ions in the original version of this note.

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L I T E R A T U R E C I T E D

Yu. L. Ershov, "Solvability of the e lementary theory of d iscre te s t ruc tures with relat ive comple- ments and theory of f i l t e r s , " Algebra i Logika, 3, No. 3, 17-38 (1964). L. Pacholski , "On countable universa l Boolean a lgebras and compact c lasses of mode l s , " Fund~ Math. , 7__88, No. 1, 43-60 (1973).

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