algebraic closure operators and strong amalgamation bases

ALGEBRAIC CLOSURE OPERATORS AND

STRONG AMALGAMATION BASES

89

PAUL C. EKLOF i)

Abstract

Axioms for a closure operator are given and it is proved that, for suitable classes, a unique largest closure operator exists. Necessary and sufficient conditions in terms of closure properties are given for structures to be strong amalgamation bases.

Introduction

In this paper we solve a problem raised by Paul Bacsich in [11, where he defined the notion of a strong amalgamation base (SAB) and proved that if K is the class of models of a universal theory with the amalgamation property then every SAB is algebraically closed in the sense of Jonsson [3] and Robinson [5]. He raised the question of whether the converse is true and proved it in certain cases. In w we prove the converse in general. Moreover we establish necessary conditions for members of K to be an SAB, even when K is not an elementary class. These conditions are ex- pressed in terms of a strong notion of algebraically closed described in w of the paper (a notion which coincides with the Jonsson-Robinson notion for an elementary class but not necessarily otherwise).

The first part of this paper deals with the problem of defining notions of algebraic closure. Various definitions of algebraic closedness have been advanced in the litera- ture, among them ones given by Jonsson [3] and Robinson [51, which have been shown to coincide (under certain general hypotheses - see [2]). In many important cases (e.g. the theory of groups) the Jonsson-Robinson notion becomes trivial. Recently Robinson [6] has introduced another notion of algebraic closedness - which, for example, is nontrivial in the case of groups. This notion of algebraic closedness does not carry with it a corresponding notion of algebraic closure, i.e. of an algebraic, algebraically closed extension. (The earlier Jonsson-Robinson papers (of [31 and [51) do define a notion of algebraic extension but a proof of the existence of an algebraic closure requires an additional hypothesis like (I4) of [31). In this paper we take an axiomatic approach to the problem of defining a notion of algebraic closure in a class of structures K (satisfying certain properties - see w 1 - but not

i) Research supported by NSF GP 34091X

Presented by B. Jdnsson. Received July 30, 1973. Accepted for publication in final form November 7, 1973.

90 PAUL C. EKLOF ALGEBRA LINI.V.

necessarily an elementary class). Our approach is in somewhat the same spirit as that of Jonsson in [4] but is different in detail. We list some properties which might reasonably be required of a good notion of the operation of taking the algebraic closure of structures in K. These become the axioms for what we call a closure operator. The principal result of w 1 is that there is a largest closure opera tor on K which is unique (up to a certain equivalence relation). In the case of an elementary class - but not in general - this largest closure operator is the operation of taking the Jonsson algebraic closure (in the sense of [3]). This then constitutes another formal- ization of the remarks of w 11 of [-3] on the uniqueness of Jonsson's notion (see also [4], w and indicates, for example, that for the theory of groups no stronger notion of algebraic closure than Jonsson's can be given - at least none which satisfies our

axioms. In the last section of this paper we exhibit some examples including one which

answers a question of Jonsson in [3].

w 1. Closure operators

Throughout the paper K will denote a class of structures of a fixed similarity type,

such that (Il) K is closed under isomorphism; (I2) K is closed under substructures; (I3) K is closed under unions of chains; (II) K has the amalgamation property.

(Compare [3], Introduction.) If A, B~K and C is a common substructure of A and B, the notation A-~ c B means there is a isomorphism of A onto B which is the identity on C. f : A ~ B is an C-embedding if it is an embedding which is the identity on C.

Unless otherwise specified all structures mentioned are assumed to be structures inK. A function F : K ~ K will be called a closure operator on K if it satisfies the fol-

lowing conditions for all A, B in K: (i) A~_F(A);

(ii) r ( r ( ~ ) ) = r ( A ) ; (iii) Every embedding f : A --* B extends to an embedding f : F (A) ~ F (B) such that

f (r(A))%~Af ( f (A)); and (iv) IfA~_Ai~_B and Ai~-a F(A) for i=1 , 2, then A t = A 2. We shall say thatA is F-closed if F(A)=A. If b~B~_A we say b is F-close to A

if F(A(b))~aF(A). (A(b) denotes the substructure of B generated by A and b). I f A _ B, we say B is F-close to A if every element of B is F-close to A.

We first list some elementary consequences of our definitions.

1.0 LEMMA. Let F be a closure operator on K and A, B, CEK.

Vol. 4, 1974 ALGEBRAIC CLOSURE OPERATORS 91

(1) I f A ~- B, then A is F-closed if and only if B is F-closed. (2) I f A~_B and b~B, b is F-close to ,4 if and only i f there is an A-embedding:

A (b) ~ F (A); (3) I fbeF(A) , then b is F-close to A. (4) I f A~_B~_C and c~C such that c is F-close to A, then c is F-close to B; (5) I f A=_B, B is F-close to ,4 if and only if r(,4)~- ~ F(B). (6) l f A ~ B ~_ C and B is F-close to A and C is E-close to B then C is F-close to A. (7) A is F-closed i f and only if Ac_B, b~B and b is F-close to A implies b~A. Proof We shall give just the proofs which are not completely straightforward. (2) If there is an A-embedding f : A ( b ) ~ F ( A ) , then by (iii) it extends to

f: F (A (b)) ~ FF (A). We claim f is onto; if not, then if g is an embedding: F (A) (A (b)), f g (F (A)) ~ F(A) and f g (i" (A)) ~- a F (A), which contradicts (IV).

(4) We have A~_Bc_B(c)c_F(B(c)). Now by (iii) there exists an A-embedding f :F (A)~F(B(c ) ) ; an A(c)-embedding g:F(A(c))---.F(B(c)); a B-embedding h: F (B) ~ F (B (c)); and an A-embedding k: F (A) --+ F (B). Then by (iv), h (k (F (A))) = =f(F(A)) . Also by hypothesis, F(A(c))~-aF(A), so by (iv), g(F(A(c)))= =f(F(A)) . Hence ceh(F(B)) and since Bc_h(F(B)), we have B(c)~_h(F(B))~- n

B F (B). Therefore by (2), c is F-close to B. (5) Suppose B is F-close to A and consider an A-embedding f : F ( A ) ~ F (B). We

claim t h a t f i s onto. It suffices to prove Bc_f(F(A)). But for any b~B we have an A(b)-embedding g:F(A(b))--*F(B). By hypothesis F ( A ( b ) ) ~ a F ( A ) s o by (iv), f ( r (A) )=g(F(A(b) ) ) Hence b~f(F(A)).

If F, and Fz are closure operators on K we shall write F~ c_F2 if for every A~K, there is an A-embedding: F I ( A ) ~ F 2 (A). We say F~ is equivalent to F2 if for all A~K, F, (A)~- a F~(A).

1.1 LEMMA. I f F1 and F 2 are closure operators on K such that Fl c_F2 and F2 c_F1, then F1 is equivalent to F2.

Proof Fix A~K. Since F2-~FI there exists a F 1 (A)-embedding f :F2 (FI (A)) Thus F2 (F, (A))= F, (A), so r (A) is r2-closed. Since

F~ __ F2 there exists an A-embedding g: F1 (A) ~ Fz (A). By 1.0(3), F2 (A) is F2-close to A, and thus, by 1.0(4), Fz (A) is F2-close to g(r , (,4)). Moreover by 1.0(1), g(Ft (A)) is r2-closed. Hence by 1.0(7), g(Ft (A))=F2 (A) and therefore F1 ( A ) - a F 2 (A).

We shall prove that on any Kthere is a largest closure operator FK which is unique up to equivalence.

There is clearly a smallest closure operator, namely the identity map I: K ~ K, for by (i), I~_ F for any closure operator F.

Let us define an operator J : K ~ K by J ( A ) = a 'maximal algebraic extension of A' in the sense of I-3]. By [3; 8.5] J(A) exists. However it is not always the case that J is a closure operator in our sense; in particular it is not necessarily true that JJ(A)=

92 PAUL C. EKLOF ALGEBRA tr~rv.

=A - see Proposition 3.3. If K is an elementary class - or satisfies (I4) of [3-] - then the results of [-3] imply that J is a closure operator. In this case note that b is J-close to A if and only i fb is algebraic over A in the sense of l-3-] and A is J-closed if and only if A is algebraically closed in the sense of 13]. We shall see that J is the largest closure operator F r if K is an elementary class. Even the weaker hypothesis (I4) does not imply that J = F r as shown by an example in w (Proposition 3.2).

IfA~_B~, b~eB~, i=1 , 2, we say b 1 is ~1-equivalent to b 2 if A ( b l ) ~ a A (b2).

1.2 Lemma. Let F be a closure operator on K and let A, B e K such that A ~_B and b e B is F-close to/1. There is in K an extension C of A(b) such that in any extension D of C i f d is/1-equivalent to b, then de C.

Proof. Let C=F(A(b ) ) . I f d e D ~ C is such that A(cD'~ A A(b) then there is an /1-embedding:/1 (d) ~ F (A (b)) ~- A F(/1). Hence by 1.0(2), d is F-close to A and thus by 1.0(4), F-close to /1 (b). Therefore by 1.0(7), de C.

We wish to prove that there is a largest closure operator FK on K. Lemma 1.2 suggests that FK should have the property that, f o r / 1 ~ B, b eB, b is Fx-close to /1 if and only if b satisfies the conclusion of Lemma 1.2. Let us say that b is K-close to A if it does satisfy the conclusion of Lemma 1.2. The following lemma gives a useful characterization of K-closeness.

Remark. We have avoided the use of the terminology b is algebraic over A since our notion does not necessarily have the finite character usually associated with notions of algebraic. However, as we shall see, our notion specializes to that of Jonsson-algebraic when K is an elementary class.

1.3. LEMMA. b is K-close to A iff there exists a cardinal rc such that in any extension D of A there are < x elements A-equivalent to b.

Proof. (=~) If there is no such cardinal, then for any extension C of A (b) there is an extension D of A such that D contains > C a r d ( C ) elements /1-equivalent to b. Then if E is an amalgam of D and C over A, E is an extension of C such that there exists e e E - C with e/1-equivalent to b.

(~=) If b is not K-close to A, then we can define by induction a chain of extensions Cv o f /1 (b) such~that for each v, there exists cveC~+l-C~ such that c, is A- equivalent to b. Then for any cardinal x, C~ contains x elements /1-equivalent to b.

If beB~_/1 is K-close to A, we shall denote by Xa,b the smallest cardinal x such that every extension of A has < x elements/1-equivalent to b.

I f A G B, we say B is K-close to/1 if every element of B is K-close to A. Let xa, a = =sup(xa ,b :beB} .

1.4 LEMMA. (1) I f every element of U is K-close to A then A (U) is a K-close extension o f A ; (2) I f A ~_B the elements in B which are K-close to A form a substructure orB.


The proof of .1.4 uses 1.3 and is analogous to the proof of the corresponding assertions in ['3] so we shall not give it here. However the following result is not necessarily true for Jonsson's notion of algebraic unless K is an elementary class (see 3.3).

1.5 LEMMA. I f A~_B~_C and C is K-close to B and B is K-close to A, then C is K-close to A.

Proof. Let csC. If c is not K-close to A then for any 2 there is an extension D of A such that there are 2 different A-embeddings fv: A (c) ~ D, v < 2. Choose 2 to be a regular cardinal > (xA, B)IBI+ x~, ~+N;o. Because K satisfies (II) there exists an extension E of D such that each f~, v<2, extends to an embedding g~:B(c)~E (cf. 3.2 of [3]). Now since 2> (xA, B) 181 there are <2 A-embeddings : B ~ E. Hence there is a # < 2 such that {v:g, rB=g, IB } has cardinality 2. But this contradicts the fact that 2> xn,~.

We shall say that B is a maximal K-close extension of A if B is K-close to A and there is no proper extension C of B such that C is K-close to A. A structure A is said to be K-closed if it has no proper K-close extensions.

1.6 LEMMA. (1) I f B is a maximal K-close extension of A, then B is K-closed; (2) if B is a maximal K-close extension of A then for any C ~_ A such that C is K-close to A, there is an A-embedding: C--* B.

Proof. (1) follows immediately from Lemma 1.5. As for (2), if B is a maximal K-close extension of A and C is K-close to A, then by (II) there exists D ~ B and an A-embeddingf:C--* D. By 1.4(1), B( f ( c ) ) is K-close to A and hence, f(C)~_B.

1.7 LEMMA. For any A~K there exists a maximal K-close extension of A which is unique up to isomorphism over A.

Proof Suppose false. Let {Bi:i~I} be an indexing of all (up to isomorphism over A) the K-close extensions of A of the form A (b) (i.e. simple extensions); say Bi = A (bl). Let 2 be a regular cardinal > [II+ sup { tea,b, :i~I}. We can define by induction on v<~2, extensions C~ of A such that for all v<,~:Cv=C~+l; C~+x=Cv(e~) for some c~s Cv+ 1; and C~ is K-close to A. In fact if v is a limit ordinal, let C, = I--Jr < ~ Cu; and if v=/~ + 1, let C~ be any proper simple extension of C, which is K-close to A. (C~ exists because C, is not a maximal K-close extension of A). By choice of 2 there exists io EI such that { v < 2:A (c~)-~ a Bio} has cardinality 2. But then there are ~. elements in C~ which are A-equivalent to b~o, a contradiction. Hence A has a maximal K-close extension. If B1 and B z are both maximal K-close extensions of A then by 1.6(2) there is an A-embeddingf:B1 "-*B2. N o w f ( B 1 ) is K-closed since B1 is K-closed by 1.6(1); and B2 is a K-close extension of A and hence of f (B1) . Thus f (B1)=B2 so B l ~ a

B2.

94 PAUL C. EKLOF ALGEBRA UNIV.

1.8 THEOREM. There exists a unique largest closure operator on K i.e. a closure operator Fr such that:

(a) for any closure operator F on K, F ~ Fr; and (b) i f F o is any other closure operator satisfying (a), then F o is equivalent to

FK. Moreover, Fr is characterized by the property that for b ~ B ~ A , b is Fx-close to A

if and only i f b is K-close to A. Proof Define F~ (A) to be a maximal K-close extension of A, which exists by 1.7.

Then F r satisfies (ii) by 1.6(1). As for (iii), i f f :A ~ B is an embedding then by (II), there exists C ~ B and an embedding g: Fr (A) ~ C extending f ; B (g (F r (A))) is K- close to B so by 1.6(2) there exists a B-embedding h : B ( g ( F r ( A ) ) ) o F ( B ). Then f = h og:CK (A)--, B is an embedding extending f Moreover f ( F r (A)) is K-close to f ( A ) since F r (A) is K-close to A ; andf(FK (A)) is K-closed because Ft~ (A) is K-closed. Hence by 1.7, f (Fr (A) )~- . c (a )F~( f (A) ) . Finally F r satisfies (iv), because if A _ A ~ _~ B and At ~- a Fr (A), i = I, 2, then A ~ is K-close to A; hence A 1 (A 2 ) is K-close to A and since A1 is K-closed, A1 (A2)---A,; so A2~_AI; similarly AI~_A z. Therefore F~ is a closure operator. If F is any other closure operator, then by 1.2 and 1.6(2), F~_F K. The uniqueness of Fr follows from Lemma 1.1. Finally, if F is a closure operator such that for any b s B ~ A , b is F-close to A if and only i fb is K-close to A, then by 1.0(3), for any A, F(A) is K-close to A and by 1.0(7) and (ii), F(A) is K-closed. Hence F (A) is a maximal K-close extension of A, so r,,(A).

1.9 COROLLARY. l f K is an elementary class then the largest closure operator on K is J, the oronsson operator.

Proof This is a consequence of the last statement of Theorem 1.8, Lemma t.3 and Theorem I 1.1 of [-3].

In w we shall give an example of a class K such that F r # J . Remark. The results of this section depend very heavily on the use of property

(iv) of a closure operator. It may be interesting to investigate what we can say in general about 'closure operators' which satisfy only (i)-(iii) and, perhaps, some weakened version of (iv). For example, if K is the class of modules over a fixed ring the operation F (M) =injective envelope of M satisfies (i)--(iii) but not (iv); it also satisfies the following 'minimality' condition:

(iv') I f A~_B~_F(A) and F ( B ) = B then B=F(A) .

w 2. Strong amalgamation bases

We say that A is a strong amalgamation base (SAB) of K if whenever A_Bi , i= 1, 2, there exists C~_Bz and an A-embeddingf:Bt --* C such t h a t f ( B 1 ) n B z = A . Bacsich proves in ['1] that if A is a strong amalgamation base of K, then A is J-closed.


We shall strengthen this result for arbitrary classes K and prove a converse for elementary classes.

2.1. PROPOSITION. For any A ~ K the following are equivalent: (a) A is K-closed; (b) Whenever A ~ B i, i=1 , 2, then for any b ~ B I - A there exists C~_B z and an

A-embedding f: BI --* C such that f (b)r (c) Whenever A ~ B i, i = 1, 2, then for any b ~,..., b, ~ B1 - A there exists C D_ B z

and an A-embedding f: B t --* C such that f (b j)q~Bz for j = 1,..., n. Proof The equivalence of (a) and (b) is an easy consequence of the definition of

K-closed. Since (c) =,- (b) is trivial it remains to prove (b) =~ (c). We prove it by induction on n. Given b~,..., b , ~ B ~ - A define a relation < by: b~-Kb i if and only if b~ is K-close to A (bi). This relation is reflexive and transitive but not anti-symmetric. Let k be such that ba is 'minimal' with respect to this relation i.e. for all i, if bi~(bk then bk<b~. Renumbering, we may assume k = 1. Replacing B i by FK(Bi), we may assume Bj is K-closed ( j= 1, 2). By (b) there is an A-embeddingf:A(b~)-* C where C~_B z andf(b~)r There exists an extension of f ,

y: (A (b,)) (c).

Also there is an A (b~)-embedding

Let h=fo (g- ' ): H = g ( F r (A (b~)) the commutative diagram

Fr (C). Then h is an extension o f f Thus we have

n 1

ul h

H - , ( c )

ul uI A (b l ) - , C ul Sul A ___ B 2

We will be able to finish the proof by induction if we can show that for any bi~H, h(bl)r 2. But if bi~H, then b i is K-close to A(bl) so bi-<bl. The minimality of b t then implies bl "<bi i.e. bl is K-close to A (bl). Hence h (bl) is K-close to A (h (bl)). Thus if h(b~)~B2, h(bl) is K-close to B2, which is a contradiction of the fact that

h (bl)=f(bl)q~B2 and B 2 is K-closed. Remark. The definition of the relation -< in the above proof is necessitated by the

96 PAUL C. EKLOF ALGEBRA UNIV.

fact that K may not satisfy the exchange property: if b is not K-close to A but b is K-close to A (c), then c is K-close to A (b). We give an example in w of an elementary class K which does not satisfy this property.

2.2 COROLLARY. I f A is a strong amalgamation base for K then A is K-closed. Proof I f A is a strong amalgamation base for K then it obviously satisfies 2.1 (b). Remark. An example in w shows that we may have, for K non-elementary, a

structure in K which is Jonsson algebraically closed but not a SAB. If K is an elementary class, the converse to 2.2 is true:

2.3 THEOREM. l f K is an elementary class and A is K-closed (i.e. algebraically closed in the sense of Jonsson [3]) then A is a strong amalgamation base for K.

Proof Suppose A _~ B t, i-=-- 1, 2. Let I be the set of finite subsets of B I --A. For each i e I there is, by 2.1(c), an A-embeddingf~:B t -~Ci such that Bzc_C t and for each

b~i, f j(b)r For each iEI let S t= {jeI: f j (b)r for all beI}. Then {St:i~I} has the finite

intersection property, since il u ... u i ,~St, n ... c~ St.. Hence there is an ultrafilter U on I such that St~U for all i. Let c=I-L~ t c i /u , the ultraproduct of the C{s with respect to U. C~K since K is closed under ultraproducts. There is a canonical embedding g : B 2 ~ C since B2gCt for all iaL Also the f{s induce an embedding f :B l ~ C. By our choice of U, for each b ~ B - A , f ( b ) r since for each deB2, {i~I:fi (b) ~ d} ~_ Sob ~ E U.

We have not been able to settle whether the converse to 2.2 holds in general. In addition to the above, we do have a converse under the following circumstance:

2.4 THEOREM. Let K be a class closed under products. I f A ~K is K-closed then A is a strong amalgamation base for K.

Proof Suppose A~_Bt, i= 1, 2. For each b~B 1 - A , let C~b~,f~b~ be defined as in the proof of 2.3 and let C = 1-Ib ~ n , - a CCb~. Let f : B 1 ~ C be the embedding induced by the fcb~'s and let g: Be ~ C be the diagonal embedding. Then f ( B l ) c~ g (B2) = A. Thus we have proved that A is a SAB.

Remark. Under the hypothesis of 2.4 it may be proved (as in [2; 3.1]) that K- closed equals algebraically closed in the sense of 13].

w Examples

We first give an example of an elementary class K which does not satisfy the 'exchange property' referred to in the remark following the proof of 2.1. Let T consist of the universal closures of the following formulas (where G and Ware unary predicate symbols; and R is a binary predicate symbol):


a. c(x) ~ w(x) 2. R(x,y)+-~R(y, x) 3. R(x , y ) ~ [(G(x) A W(y)) v (G(y)A W(x))] 4. [G (x) ^ R (x, y) A R (x, z)] ~ (y = z)

Informally, a model of Tconsists of two disjoint sets of elements ('greens' and 'whites') and a symmetrical relation between elements of different 'colors' such that each 'green' is related to at most one 'white.'

3.1 PROPOSITION. I f K is the class of models ofT, K satisfies (I~), (I2), (I3) and (II). Moreover there are models A c_ B of K and b, c~ B such that b is not K-close to A, b is K-close to A (c) and c is not K-close to B.

Proof Since T is a set of universal sentences K clearly satisfies (Ix), (I2) and (13). We shall not give the straightforward proof that K satisfies (II). Let B = (a, b, c}, A = (a} where G (a), W(b), G (c), R (b, c), R (c, b) and no other relations hold. Then b is not K-close to A (because ,,~R(b, a)). But b is K-close to A (c) (because R(b, c ) ^ ^ W(b)). However c is not K-close to A (b) (because G (c)).

Next we give a simple example of a class K t (necessarily non-elementary) where Kt-close does not coincide with Jonsson algebraic. Let Kt consist of all structures 9~ = (A, U, < , S, P ) where < is a discrete linear ordering on U and S and P are unary functions such that either U= 0 or (U, ~<, S, P ) ~ (Z, ~<, s, p), the integers with the usual ordering and successor and predecessor functions. Note that if 9~___92'= (A ' , U', '< , S ' , P ' ) and 9~' ~KI, then either U=0 or U= U'.

3.2 PROPOSITION. K1 is a class satisfying (I,), (I2), (13) and (I1). There exist members o f K1 which are algebraically closed in the sense of Jonsson [3] but which are not Kt-closed (and hence are not strong amalgamation bases).

Proof It is easily checked that K~ satisfies (I~), (I2), (13) and (II). (K 1 even satisfies property (I ,) of [3]). Every member of K~ is algebraically closed in the sense of Jonsson. But if 9~K~ such that U = 0 and 92_c92'eK~ where U ' # 0 then i f a e U', a is K~-close to 9~ by Lemma 1.3, since there are only a countable number of elements 9~-equivalent to a in any extension in Kt of 93[. Note also that 9~K~ is K~-closed if and only if U#0. Also 9.I~K1 is Kt-closed if and only if 9.I is a strong amalgamation base.

We now consider another example in which K-closed is not the same as Jonsson algebraic. In this example (14) fails and in fact so does 9.2 of [3], answering a question of Jonsson. Let K 2 be the class of all structures 9~= (A, F", ..., U, •'),<o, where A = U w [,.J, F" (disjoint union); F" is a field (and the field operations on F" are given in ... such that any substructure of (F", . . .) is a field); U is a subset of I-[,<,~ F"; and 7:": U ~ F" is the canonical projection (i.e. re" ( f ) = f (n) for f~ U). By J-algebraic we shall mean algebraic in the sense of [3].

98 PAUL C. EKLOF

3.3 PROPOSITION. /(2 is a class satisfying (Ix), (I2) , (Ia) and (II). There exist structures 9ii in K 2 ( i=0 , 1, 2) such that 9~o~911 ~912, 91 is J-algebraic over 9~o, 912

is J.algebraic over 9~ but 9~ 2 is not J-algebraic over 9.[ o. Also, every max ima l J-algebraic extension o f g~ o is not J-algebraically closed.

Proof. We omit the easy verification that K 2 satisfies ([l), (I2), (I3) and (II). Observe that if 9/has the form described above and 9~_ 9~ 1 = (At , F~', ..., Ua, ~c~'), <o, and a~A 1, a is J-algebraic over ~2[ if and only i fa~F~ for some n and a is algebraic over F" or a e UI and rc~' (a) is algebraic over F" for all n<co and rt~' (a)~F" for almost all n < co. Now let F~ = Q, F~' = Q (x/2) = F~ for all n < co. Let Uo = 1--[, F~ ----- U1, U2 = I'I, F~. Finally for i=0 , 1,2 let A , = U i w [,_),F 7, and 9~,=(A, ,FT, . . . , U,, n~.),<,~. Then ~o__q~t_~2 ; By the above observation, 9~ is J-algebraic over 9,I o and 9~ 2 is J-algebraic over 9d t but 2[ 2 is not J-algebraic over 9/o. A maximal algebraic extension o f ~ o must be of the form 9 i . = (A. , F . . . . . . U., f t . ) where Fg is an algebraic closure of Q and Uo= U.=I--[. F~,. But then there exist proper algebraic extensions of 9i,, for example 9~**=(A**, F~. , . . . , U**, rrg,) where F ~ . = F g and U**=I'-[. F~.

Remark. In the above example, 9~= (A, F", .... U, zr").<o, is K2-closed i f and only if each F" is algebraically closed and U=]--[.<,~ F". It is clear that 9~ is an SAB if and only i f ~ is K2-closed.

REFERENCES

[1 ] P. Bacsich, Strong amalgamation bases, preprint. [2] P. Bacsich, Defining algebraic elements, J. Symbolic Logic 38 (1973), 93-101. [3] B. J6nsson, Algebraic extensions of relational systems, Math. Scand. 1l (1962), 179-205. [4] B. J6nsson, Extensions of relational structures, The theory of models, (North Holland, Amster-

dana, 1969), 146--157. [5] A. Robinson, Introduction to model theory and the metamathenzatics of algebra (North Holland,

Amsterdam, 1965). [6] A. Robinson, On the notion of algebraic closedness for non-cormnutative groups and fields, J. Sym-

bolic Logic 36 (1971), 441 4A.A.. Stanford University

University o f California, #vine

&vine, California, 92664 U.S.A.

algebraic closure operators and strong amalgamation bases

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