4t

5,

6,

7.

N. V. Beiyakin and L. N. Pobedin, "Dialog aspects in the foundations of mathematics," All-Union Conf. on Applied Logic, Lecture Theses, Novosibirsk (1985), pp. 17-18. E. G. Hikiforova, "A method of constructing enumerations, Mathematical logic founcations of the MOZ problem," Vych. Sistemy, I07, 80-95 (1985). E. G° Nikiforova, "On a class of recursive hierarchies," All-Union Conf. on Applied Logic, Lecture These, Novosibirsk (1985), pp. 164-165. S, C. Kleene, "Recursive functionals and quantifiers of finite types," Trans. Am. Math. Soct, 91, No. I, 1-52 (1959).

B-DIVISIBLE ALGEBRAS

A. I. Omarov UDC 512.57

Jonsson and Tarski (see [3]) considered algebras ~ for which isomorphism of the Boolean

powers ~] ~ [~j implied isomorphism of the Boolean algebras ~i-- ~- ~g Such algebras

were called ~ -divisible, and Burris [3] first studied them systematically.

In [3] many unsolved problems were formulated, of which we are interested in the fol-

lowing:

a) Does there exist a ~ -divisible Abelian group?

b) If ~ is a finite E-divisible algebra and ~J is equationally compact, then is

B a complete Boolean algebra?

c) If ~[~ is K -saturated (K~UJ) , then is ~ a K -saturated Boolean alge-

bra?

d) If ~ is ~ -divisible and the Boolean powers ~[~lJ and ~Z] are elementarily

equivalent, then are the Boolean algebras ~I and ~2 elementarily equivalent?

These problems are considered in this article. Problem a) has a negative answer, prob-

lem b) has a negative answer for algebraic systems and a positive answer for non-Abelian

algebras, c) has a positive answer if we postulate the generalized continuum hypothesis

(GCH), and finally problem d) is partially solved.

Baldwin and McKenzie [2] applied Boolean powers to characterize the spectra of univer-

sal Horn classes (quasimanifolds), and they formulated the useful condition (A), which for

universal Horn classes is analogous to the condition of stability in complete theories. We

that the theory ~ has the property (A), if all the propositions of the form ~f~$1~C~I] say

:\~<~)A~(~,~)-~d~,~)), where @(,) runs through the atomic formulae of the language ~=~(~

with 2~ variables, belong to this theory.

The center of the algebra ~ is the binary relation

where ~ runs through all the terms of the language ~(~) and U,~£ t~l The algebra

is called Abelian, if ~,,= l~J ~ .

Translated from Algebra i Logika, Vol. 25, No. 3, pp. 315-325, May-June, 1986. Original article submitted June 17, 1985.

0002-5232/86/2503-0199512.50 ~ 1987 Plenum Publishing Corporation 199

Clearly, condition (A) implies that the algebra is Abelian. This concept turns out to

be useful in the study ~ -divisible algebras, in particular for considering problem b) and

the description of non-Abelian ~ -divisible algebras.

From the elementary classification of Boolean algebras [i] it is easily seen that a com-

plete elementary theory of Boolean algebras (a type of Boolean algebras) has a ~ -saturated

model and. an atomic model. It turns out that this property also carries over to an elementary

type of Boolean powers of a finite algebraic system.

We note [4] that the fundamental results of this article can also be carried over to

~-powers if we replace ~ -divisible algebras by ~-divisible algebras.

We shall use generally accepted definitions and notation without explanation.

THEOREM i. If ~ is a finite non-Abelian algebra, then the algebra i~loei~i is

-divisible.

Proof. Since the algebra ~ is non-Abelian, there exists an atomic formula 61~)

and a pair of strings ~,$~I~I ~ such that we have

For simplicity we shall assume that ~ and ~ are strings of length i, i.e , ~=~ and

Consider the set Then the set of

elements of the Boolean power ~3 , defined by the formula O~,2) A~(~,~)A@(2,~) , forms

a partially ordered set with respect to the predicate @(~} , isomorphic to the Boolean

a!gebra ~ Thus the Boolean algebra ~ ~is formular in ~[~] and thus I~,a)aei~ I is

-divisible. Now suppose that ~ ~ ~ J We divide $ into two subsets, setting

z ¢ S z o(o, xlA (x,o) , x c £

Let ~={~C 2 .... ,C~ be a list of the elements of S I From this list we eliminate all the

elements which do not satisfy the formula 6 (~,~) , and then we take the first noneiimina-

ted element C I after ~ and eliminate from the list ~ all the elements which do not

satisfy the formula @(~,C~2) ; we then take the first noneliminated element C~z after C 4

and eliminate from the list ~ all the elements which do not satisfy the formula ~I~,Ci ) ,

etc. After a finite number of steps the process ends, and we obtain a list of noneliminated

I ~/I C/~ ~ and a system of formulae 6~,~),~(JC, C/z%,.,,d/~F,~ ~) Consider elements ~I = , "" "'

the formula ~(~)=~,X)a~(..T,.:~)A@(..T,~)AD{.T,Q~)Ao..A~%'~ Let ~;°--{~,4,o.,,L1/~} be the complete list of elements in Sg , satisfying the formula ¢[~ We have ~ #~ , since

6~ . We now eliminate all the elements of SZ which do not satisfy the formula ~f,.r)

we then take the first noneliminated element ~7 after ~Kz in the list S; and eliminate all

the elements in S2 ° which do not satisfy the formula 6[~ X ,X} , etc. After a finite num- g

beT of steps, the process of eliminating stops, and we obtain a list of noneliminated eiem-

ents

200

and a system of formulas

Consider the formula

properties of the formula B(.~')

l)

2)

3)

4)

~ ( d ) <-----4 ~E~'U~; for each element

d,. 6 %,d, !

a e S~.

We note the following

~ I%J (by construction);

By construction, the element ~7 has the property O(L2,.~'}A'I~C2,LZ) for each x6S~ , but

in general not all elements of S; have this property.

ist case. For each element d~ ~; , ~6 SI I let us have $ ~ O~,C) A q ~,~) . Consider

the formular subset ~(~ [~3) , defined by the formula ~(X} Clearly, ~ ~I~[~])

if and only if each coordinate ~z" belongs to the set 4 f U ~; In the set ~(~[~J) , the

formula O(~)A@I~,~) defines an equivalence relation, and moreover ~he factor-set of this

relation with respect to the predicate ~(~,~# forms a Boolean algebra isomorphic to ~ .

2nd case. We shall narrow down the formula /~(X) in stages:

Step o. S,',

• -£)A6(. .~,dB¢), .~'6S °} = S °" . If $a, is empty, then set S I - ~ a , 4(.fC)=4(, ,~.) and go to the 2nd step, and if it is not empty, then set ~ ' = ~ and 4~']='._~/,.~)AO(..~,~B, ) , and go to the

2nd step.

Step 2. Take an element ~BZ and consider the set I~I@~.~A@(~TI~),3CE~IJ -~ $,t .If

this is empty, then set $2=$1 and 4[JT.) =~flx) , and if it is nonempty, then set S~=SV

4(~)=4[~)od(~,~) , etc. We perform ~ steps and obtain S=i~1,~,... , and a formula 3 z

We note the following properties of the formula ~(~J and the set SO :

2" 7 /

~e'¢u.,z) A~(x,u)aof~x)^o(~,a'9.

Write oL(,.~)U~61(l,b)aSi(~a6"~l,t,,bi~/%"16~gl, D~4~"(~l, J 4 1 ~ ( ~ a ~ a ~ . Now, in the Boolean power

U~ consider the formula ¢~)--~[~}h~(~ . For each element ~ such that ~[~] ~ 4 ~)

we divide the set of coordinates into three subsets X , Y , Z ,XLIYuZ=/ , setting X={Z

I~[~ ~ IxSO}, Y=[~'I~.~S~), Z={Z[~.£~I . Then X ,Y, and Z are elements of the Boolen algebra

We note the following property: if ~[~] ~= ~V(~) , then ~[~ ~ q~g)~-~ ~ = ¢ . In

fact, if ~#~ , then ~6~ , and for ~i, ~ , there exist ~£$~,~.£~jx S 0 such that

20:

: Taking elements ~ and ~ such that

we have ~ [~] ~ ~ ( d ) . z= fJ

exist elements /../,/.J"~ ~[~] satisfying the formula

Suppose that ~3~[~) . Then there

i.e., {;I~ ~7~,~)}=Z~ It is easily seen that this is possible if and only if 46

20 for i6: , and thus ~[~3~ ~(d) , i.e., ~[~]~ U~[~) Now, as in the first

case, it is easily seen that the formular set ~I~]}~(~/[~]) , after factorization by the

relation O(~)A~(~,2 ) , forms a partially ordered set with respect to the predicate @I.~)

isomorphic to the Boolean algebra ~ The theorem is proved.

It follows from the proof of the theorem that any finite algebraic system ~ with ~he

predicate ~(~ , f ) , satisfying the condition ~,~.~,~(~(~,~- ' )A@(~,~)A~e~)A~d(~,~)) , is $ -divisible.

However, for such a class of algebraic systems Bursis' problem [3] has a negative solu-

tion: if ~ is a ~ -divisible algebraic system and ~[~] is equationally compact, then is

the Boolean algebra ~ complete?

Proposition. Let ~=(~,O(,)> be a finite algebraic system with the predicate

(2,~) , satisfying the condition

and let there exist an element a~t~l such that ~ Vf(~(~,~)A~(~)), where ~=<n,n ..... a ~

Then ~ is a ~ -divisible system and for any Boolean algebra ~ , the system ~'~J is equa-

tionally compact.

Proof. It follows from Theorem I that the system ~ is ~ -divisible. We prove the equa-

tional compactness of ~] There exists ~ 6 l~I and ~[~] ~(~,ff)A~,~}) Let

~,~) be a locally satisfiable set of atomic formulas. Then i~ is satisfied in ~3

if we interpret all the variables ~ by the element ~Z

Although the condition (*) was essential in Theorem i, it is not equivalent to the con-

dition of being non-Abelian, since we have proved a Proposition restricted only by the condi-

tion (*), whereas a non-Abelian algebra gives a positive answer to Burris' problem [3] in the

case of algebras.

THEOREM 2. Let ~ be a non-Abelian algebra; then if the Boolean power ~/~] is

equationally compact, the Boolean algebra ~ is complete.

Proof. Since ~ is non-Abelian, there exist a term ~ and elements ~.~,~ , ff such

that ~a)--~(~)--~-~, ~(~,~)~, ~I~}=~z For simplicity, let ~=~ ,f=~ be

strings of length i. Let i~l~£~j be a sequence (a directed system) of elements of the

Boolean algebra ~ On the set ~ , consider an ultrafilter ~ , containing the directed

202

system of indices Z~¢ -- {~I~7~/" D }£ F Since

exists a retracting homomorphism ~:~[~j]i/~ --~ ~[~]

~,~(~=~}, Consider an element ~ in ~[~J~r/F

by the condition

~] is equationally compact, there

Denote by V the set [~fl #(~,~)=

such that for each i~ ~ is define(

~,,e show that x=h'la:~:?;~ v} fact, we show that ,.~"~X tion

{ a , ,f j c X ; , d,.q~-- q , if j ¢ -L~u~a "~

,4 a'~ ~ d [m], t ~ d , a ; - e .

is the supremum of the system of elements [~ l~£r} • In • Consider the element ~ ~ ~ ~] defined by the condi-

{ d~ if

Then we have /IJ(~(~,C~};=~(~(O~),~c),LIOJ5%}£/E, and

and thus ~{/~{~/}.~#=:~ Let iEX~. Then

6¢X~ .

~(/2~}=":~ where :~{~'l=I ~' i t X . , ' c:, i e X . ~ ,

and thus ~](~} £ V , i.e., X~X Now let Y be an arbitrary upper bound of the system of elements {~I~6

~ . We show that ~Y . Suppose not, i.e., Y~X Consider the element :E~[~]

d e f i n e d by the c o n d i t i o n . [ a , & ~ X " Y " Then # ( d , : ) = ~ , where ~7(l,7=~e, ,LtX~, i.e., : ,4E~[~ . ] , and thus tO4(d),p)=y Then for each &£X x W we have ~'~/~(~)(~4 f~:):7a')%=tW&c&dJ . On the o t h e r hand, by the d e f i n i t i o n of V , ,,,e see t h a t ~ ¢ X

since ~(~(d;,dffC l , and thus X- Y: d Consier now the following Burris problem [3]: is it true that if ~ is a ~ -divisible

algebra and ~[~]=~J then ~=~ ? In future we shall assume that ~ is a countable

algebraic system. Two countable Boolean algebras ~ and ~ are called connected, if for each

finite constant extension I~, g),~,~) , and an arbitrary locally satisfiable system of form-

ulae ~(~,~,,.. C} , satisfiably in one system implies satisfiabili=y in the other; moreover,

there exist systems of elements {~'16<~}~ I~l and [GiI&<UJ}C-l~I , satisfying all the form-

ulae in E(,21,~TZ,.,,C ) , such that a~=O~=~=O, ~<~

THEOREM 3. If ~[~] m ~[~] and the Boolean algebras ~ and ~ are connected, then

Proof. We apply ~he shuttle method of constructing an isomorphism. Let i) U[~]=~QI,

Qd,...,LT~,..} be the entire list of elements of ~$~J , and 2) ~==(~t, 4 .... '4'"'} be

the entire list of elements of ~[~J .

Step ~. ~et ~ X ; be the type of t~e element ~, , i.e.,~,:~:~ [~I~ ; I~ [~ 'CO;~} . Consider ~, (3~} = {o(, (~;,oCz[.~£,. } , where F ~(~)-"~ (¢) , By the Feferman--Watt theorem we

have a locally satisfiable set of formulae ~,~(~,~ .... ) in ~ and in ~ , Since Q: sa=is-

lies all the formulas in ~:(~) , then substituting the element ~2~ , we see that

203

--~ • Q, a, ~Q,I~,~;,...) in ~ is satisfied by some system of elements X I ~ .... in ~ Since

G,E~[~ 3 , then I~'I~E~/'}/<~.x.) and ~,XZ .... form partitions; then the variable ,~- , ex-

cluding a finite number of variables ~ , take the value 0 , i.e., 2~E (2,,...,Za,U,0,...,~)

Since ~ and ~ are connected, there exist YI,~ ..... ~6]~I such that ~Q(~/,...,Ym,O.0,,...~ , cor- responding with which we choose the first element in the list 2), which Satisfies all the for-

mulas in .~Q, IX) Let this be [K~

Step 2. We take a new unmarked element in 2), ~KZ Consider the type of the element

~Kz over (~, :

By the Feferman-q4att theorem, to this there corresponds a locally satisfiable set ~I~I, ~ .....

~n~,...,~) in (~,~,...,~0~) Since I~[~J,Q~)-(~[~],$~) , then the corresponding set of o, o,

formulae ~.~,...~,='~/'~/ is locally satisfiable in (~,Xf .... ~= ) as well; substituting the

element I in place of ~2 , we obtain a system of elements [ o o o y/,,.,,~, Y~, .... satisfying

all the formulas in EIZ,6,..,Y,,.%I , and moreover ~:=0 ,K>~[/Z Since the Boolean el- o, ,

~ebras ~,Xw.?~) and (~,Yt'""~) are connected, we obtain a corresponding system of ele-

ments X,,...,Xm, .... satisfying all the formulae in ,~, .... , , where ~=0 , K)~ .

For ~°,.,,,~2~ we find an element ~Z ' satisfying all the formulae in P~. [~L21) . Continuing

this mapping at even steps from ~[~] to ~j , and at odd steps from ~zf~'[~]to ~[~'I , we

obtain an isomorphism ~[~] ~-----~[~]

COROLLARY i. Two countable, connected, elementarily equivalent Boolean algebras are iso-

morphic.

Proof. Let ~ be ~ -divisible, and ~ and ~ be elementary equivalent and connected

Boolean algebras, then ~C~]---~[~3 , and since ~ is ~-divisible we have ~ .

COROLLARY 2 [3]. If ~ is a finite ~-divisible algebra, then the elementary equival-

ence ~[~]--~3 implies the elementary equivalence of the Boolean algebras ~ & .

Proof. We have ~[~]'~[~,~]-4~[& 3 where ~" and ~ , are countably saturated

Boolean algebras, such that ~-~ ~e,~.~, . Then ~[~] ,~[~3 are countahly saturated,

and thus ~--~[~ Since ~ is ~ -divisible we have ~'~--- ~ -, and thus ~m~

Theorem 4. Let ~[~] be ~9 -saturated and let ~ be ~-divisible, then ~ is ~ -

saturated. Assuming the GCH, if ~[~] is ~+-saturated, then ~ is o~ +-saturated.

Proof. Let ~[~3 be U) -saturated and let ~ be countable. Then ~[~] ~-- ~ [~lj ,

where ~t iscountably saturated, and as ~ is ~ -divisible we have ~_-- ~/ Let ~ be

uncountable. Since by the Feferman--Watt theorem types over ~[2~] are defined by types

over ~ and ~, then ~'~I~[2~]) has a ~)-saturated model. We show that ~ is ~)-sat-

urated Take ~,CZ, G ~ l~I and consider the type p(O-,C! .... ,C~) in ~ over ~,~7 .... ,~ . We

show that it can be realized where is S -saturated. Consider

such that ~----- ;~ ~,~z,~61~I,~/C~[~,] We construct ~z such that ~z' ~E~]C ~2

"< ~[~_] , and we then find ~Z_C~ such that ~ZC~[~z] , etc.

c

J&~]=~ [/J~i,] = ~JU/[ < ~ [~ ] , ~,/~/~ is a -saturated, and then ~/[U ~ ] is ~ -saturated,

and therefore ic~ ~f is ~ -saturated and the type ~I~C/,~...,~a) is realized in it.

204

General case. Let 2 ~= oC + , and let ~E~] be ~+-saturated. If ~=~+ , ther

as ~ t ~+ we take an -saturated Boolean algebra of cardinality oC ÷ Then ~[~] : ~ ~ ,

and therefore ~ i and thus ~ is o~+-saturated. If ~ >oC + , then we proceed as

C~ ~< be a type over ~ , and we show that it can be realized. Let follows: let ~(~,.o., ...

~I be the Boolean algebra generated by ~,~,.o.,~,...~y<~ . Clearly, ~,~= (~ Consider

such'that ~[~]~,~[~ , where ~z is ~-saturated. The cardinality of ~ is ~+

(this can be constructed), and we continue the process transfinitely

Let ~!~+~]~U+U~=~ ' The system U I is ~C+-saturated, since any type ~(~,~,...,C~..~

can be realized in some ~',~U ~.]~ ~[~]-->U~ ~6~', where ~' is an ~+-saturated Boolean ~+

algebra, and then the type ~(U'~G , .... C~,..)~<~ can be realized in U~

In conclusion, we show that an Abelian group is not ~-divisible, and this answers

problem 2 of [3].

Take the Boolean algebra of finite and cofinite subsets ~-- <~(~)U~(~O,~,\> , and the

algebra ~Z : we show that for any Abelian group ~ we have ~]: ~ [~ .

It is sufficient to show that G[~]~ ~(~} _ ~ where ~(~) is the direct sum of ~ cop-

ies of the group ~ . We construct a mapping ~:~[~._,~(~u[0J) . Let ~E~[~] , then

there exists a finite number of elements OI,Q2,...,~K, fE~ such that for each ~< ~ we

have ~E[O~ .... ~K.~J and {/ I~'=~S} is the union of a finite number of atoms, and {jI~' {j

are cofinite. We then set ~(~)=<~,~ ..... ~K>, where 4=~, ~=Q~...,~=~-~ This mapping

is an isomorphism. We note that ~ ~ , since there exists an element ~z in ~ ,

equal to the union of two elements ~ and ~z such that Q~2= ~ , and each of these con-

tains an infinite number of atoms.

2,

3. 4.

LITERATURE CITED

Yu° L, Ershov, Problems of Solubility and Constructive Models [in Russian], Nauka, Mos- cow (1980). J, Baldwin and R. McKenzie, "Counting models in universal Horn classes," Alg° Univ., 15, 359-384 (1982). S. Burris, "Boolean powers," Alg. Univ., 5, 341-360 (1975). A. I. Omarov, "The elementary theory of ~ -powers," Algebra Logika, 23, No. 5, 530- 537 (1984).

205