Ob Algebraiceskih Konstrukciah Sohranauscih Kompaktnost. by S. R. KogalovskijReview by: Leszek PacholskiThe Journal of Symbolic Logic, Vol. 39, No. 2 (Jun., 1974), pp. 338-339Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272669 .

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338 REVIEWS

The following, although not the most general result obtained, gives a flavor of the kind of interpolation theorems that are proved in the paper under review: Let f(R, x, y) and X(S, x, y) be formulas of the first-order languages L(R) and L(S), respectively. Then the following three conditions are equivalent: (i) F 3yVRVSVx[i - xl, (ij) there are formulas a(y) and O(x, y) of L such that k 3ya and k VxVy[(a A b --> 0) A (O X)], and (iij) there is a formula +(x, y) of L such that k 3yVRVSVx([k -> A] A -O X]).

The principal result concerning definability is the following: For any first-order theory T in a language L(P) and any positive natural number n the following two conditions are equiv- alent: (i) For every structure 21, I{P c A : (21, P) k T}l _ n, (ij) there are formulas a(V1,

- *, Vk) and Oi(x, v1, * *, Vk), 1 < i < n, of L such that T I v** *Vka and T k

Vz'1 ... V*k*a -e- V1:5t5Vx(Px 4-+ Oil]. The definability results are obtained with the help of the interpolation results and the latter

are proved by use of the following theorem about special models (for definition of special models see Morley and Vaught's XXXII 535): Let 9(R, y) be a formula of the first-order language L(R). Then there are formulas Oj(y) of L(i E I) such that for every special model 21, 21 Vy[3RO(R, y) Ai-, &(Y)]*

The results about definability and interpolation are improvements of well-known results; for example, Beth's definability theorem is obtained from the quoted result by setting n = 1.

The lemma about special models is also used to obtain some results about the intersection of elementary submodels. In particular, Park's theorem characterizing the subsets of the universe closed under a certain kind of definability.

The style and balance of the paper make it easy to read. The only criticism that could be made (in addition to a confusion of German 21's with script s's) is that no applications are made of the interpolation and definability results obtained; on the other hand it can be argued that the latter is in the domain of the reader. E. G. K. L6PEZ-ESCOBAR

ANDRZEJ EHRENFEUCHT. Elementary theories with models without automorphisms. The theory of models, Proceedings of the 1963 International Symposium at Berkeley, edited by J. W. Addison, Leon Henkin, and Alfred Tarski, Studies in logic and the foundations of mathe- matics, North-Holland Publishing Company, Amsterdam 1965, pp. 70-76.

The r-spectrum K(T) of an elementary theory T is the class of cardinals K such that T has a model of cardinality K for which the identity is the only automorphism; such structures are called rigid. For each countable ordinal a, the author exhibits a countable theory Ta such that K(Ta) = {KT: No _ K < '-Ka For each ordinal a _ M,6, a slight modification yields a theory Ta, with fewer than M,6 non-logical constants, such that K(Ta) = {K: No< K K< XK}.

The crux of the construction is the fact that standard models of set theory are rigid. The author defines a theory T* with infinitely many finite models, such that each finite model is rigid and no infinite model is rigid; for each countable ordinal a, there is a countable extension of T* which defines a well-ordering 0 of the finite models of T*, the type of 0 being cw + a. The ordering 0 is used to index the levels of an extensional relation E; since 0 is a well-ordering, the relation E is isomorphic to the standard membership relation on a subset U of V<,+ a (V<,+a, is the set of sets of rank <W + a). If U = Vc,+ a, the resulting model has cardinality Ka (assuming the generalized continuum hypothesis), and elementary submodels provide rigid models of all smaller cardinalities.

The author poses two open questions concerning r-spectra. In fact, little is known about the structure of r-spectra; e.g. it is not known whether K, p c K(T) and K < A < p implies A e K(T). WILLIAM GLASSMIRE

S. R. KOGALOVSKIJ. Ob algibrai6eskih konstrukcidh sohrandfikih kompaktnost'. Russian original of XXXIX 368(1). Sibirskij matemati&skij Zurnal, vol. 8 (1967), Pp. 1202-1205.

Let -W = {2i : i e I) be a sequence of relational structures of a given language L. For a predi- cate P of L let Kp = {(i, xl, , Xm) 521t A P(x1(i), * * *, Xm(i))), and let L* denote the language {Kp: P e L}. Let II be a set of formulas of L* which are of the form Q3i1 ... Qnin4P(i , - - *, in , X1 , * * * Xm) where cp is open and Q is V or 3. Finally let IIW denote the structure <9EIAi, R,,>0En where 91 denotes the product operation, Ai is the universe of 2tl, and Rcp i {(xl,

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REVIEWS 339

Xn): <Pi.eA1 J I, KP> PEL k p(x , , x4)'. The structure 11.sl is called a regular product. The notion of regular product was introduced by Mal'cdv as a generalization of the direct product operation. If K is a class of relational structures, then 11(K) = - I'z :1 - {Ztt: i e 2-LE K} and J1*(K) = {1ld:&l = = {A: ie I,, 21E eK and 21i = 2lj for i,j eI}.

In the paper Kogalovskij proves that regular products and regular powers preserve compact- ness, i.e., for every II and every compact class K the classes 11(K) and fI*(K) are compact. This generalizes a theorem of Makkai on direct products. He also generalizes a theorem of Vaught on substructures of direct products, namely he proves that if K is closed under ultraproducts, then the class SII(K) (of substructures of elements of I1(K)) is axiomatizable by universal sentences. The same is true if we replace 11 by rI*. LESZEK PACHOLSKI

S. R. KOGALOVSKIJ. 0 kompaktnyh kiassal algebraijcskih sistem. Russian original of XXXIX 367(24). Algebra i logika, vol. 7 no. 2 (1968), pp. 27-41.

By an operation the author means a function which to every sequence of similar relational structures assigns a relational structure. If F is an operation and .-V a sequence of L-structures, then the similarity type of F(ag) depends on L and F only and is denoted by F(L). If F = {Fe: e E E} and G = {Ge: e e E} are sequences of operations then the pair (F, G) is consistent if for every language L and for every formula a of G(L) there is a formula a* of F(L) such that for every sequence z/ of relational structures for L and for every e E E, Ge(jV) k a if and only if FeC(W) k a*. Let 6 = <S, u, m, -, 0, 1, M./,<, be a generalized algebra of subsets of I (in the sense of Feferman and Vaught). Then Fa denotes the operation which to a sequence

= {At: iE } assigns the algebra <S, u, rA, -, 0, 1, Mn, Ke>n<\,9 St(L(A)), where St(L(A)) is the set of all sentences of L(A) and Ke = {i E I: 2-1 = fi. By 9Z the author denotes the generalized product operation of Feferman and Vaught with 6 as the basic Boolean algebra. If K is a class of relational structures, and F is an operation, then F(K) = {F(&<V) :!i = {(24 i 1iE, O1t E K} and F*(K) = {F(./) E F(K): 2ti = 2t1. If F= {Fe: ee El, then F(K) = U{Fe(K): e E E,. F preserves compactness if and only if for every compact class K, the class F(K) is compact. In the paper Kogalovskij investigates the question whether the generalized products of Feferman and Vaught preserve compactness. He uses a consistent pair of operations to reduce the question about generalized products to the question whether or not a certain simple operation preserves compactness. If ?1 there are some general remarks on the method. The main results are contained in ?2 and ?3. Theorem 2 says that if (F, G) is a consistent pair of operations and F preserves compactness, then G preserves compactness. Also (Theoremn 3) if S is a compact class of generalized subset algebras and F, = {Fee: Z e SI, then Fs* preserves compactness. As a consequence of these results the author gives the main theorem of the paper (Theorem 4) which says that preserves compactness provided S is a compact class of algebras. This theorem generalizes earlier theorems by Makkai, Omarov, and the ' -thor. In Theorem 4 .* cannot be replaced by .Y? A counterexample is given in ?3. How- ever if Z0 is the class of all subset algebras without additional relations, then 4'0 preserves compactness. In ?5 Kogalovskij uses the method of consistent pairs to obtain a generalization of a theorem of Vaught on direct products. LESZEK PACHOLSKI

ABRAHAM ROBINSON. Algebraic function fields and non-standard arithmetic. Contributions to non-standard analysis, edited by W. A. J. Luxemburg and A. Robinson, Studies in logic and the foundations of mathematics, vol. 69, North-Holland Publishing Company, Amsterdam and London 1972, pp. 1-14.

The author studies valuations on certain subfields of a given non-standard model of a finite algebraic extension of Q. More specifically, let K be a finite algebraic extension of Q, *K a non-standard model of K, and A an algebraic function field over K which is at the same time a subfield of *K. There are three kinds of internal valuations on *K and under certain conditions of non-triviality (in the non-standard sense) on A, they induce valuations of A over K. Moreover all valuations of A over K can be obtained in this way.

Such algebraic function fields A can be obtained by considering curves 1 given by an equa- tionf(x, y) = 0 wherefe K[X, Y]. If(a, g) E *K2 is a non-standard point on *, then K(c, ,) c *K is an algebraic function field over K. It is shown that iff is absolutely irreducible, then *1 contains a non-standard point (a, g) if and only if I contains infinitely many points (a, b) E K2.

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