Classifying ?0-categorical theories

GEORGE Classifying N0-Categorical Theories W E A V E R

Abstract. Among the complete No-categorical theories with finite non-logical vocabularies, we distinguish three classes. The classification is obtained by looking at the number of bound variables needed to isolated complete types. In class I theories, all types are isolated by quantifier free formulas; in class II theories, there is a least m, greater than zero, s.t. all types are isolated by formulas in no more than m bound variables: and in class III theories, for each m there is a type which cannot be isolated in m or fewer bound variables. Class II theories are further subclassified according to whether or not they can be extended to class I theories by the addition of finitely many new predicates. Alternative characterizations are given in terms -of quantifier elimination and homogeneous models. It is shown that for each prime p, the theory of infinite Abelian groups all of whose elements are of order p is class 1 when formulated in functional constants, and class III when formulated in relational constants.

This paper presents a classification of complete, No-categorical theories in those first order languages whose sets of non-logical constants are finite. This classification is anticipated in Clark and Krauss [7]. Following Weaver [21], for ~ a model of T, A " / d , n + k denotes the partit ion of A" generated by finitary Scott formulas in k distinct bound variables. Each complete theory T is associated with the function PT, called the partition function for T. PT is defined on c0 and Pr(n) = n + k provided k is the least natural number s.t. A " / s l , n + k = A " / d , n + l, for all I > k; otherwise PT(n) = ¢0. PT.(n)eo provided all complete n-types of T are principal; and PT(n) = n + k provided k is the smallest number of distinct bound variables need to isolate the complete n-types of T. When T is N0-categorical, the action of the partit ion function indicates which partial isomorphisms between countable models of T can be extended to isomorphisms (Theorem 1.3); and, provides a measure of the complexity of the countable models of T (Lemma 2.1).

T is class I provided P~(n) = n, for all n; T is class II provided there is k >I 0 and l > 0 s.t. for all n >t k, PT(n) = n + l; and T is class III provided for each l there is n s.t. n + l < P.r(n). Class I theories are exactly those whose countable models are ultrahomogeneous; class II theories are exactly those whose countable models are relatively homogeneous but not ultrahomogeneous. F rom results of Clark and Krauss [7] it follows that class I theories are exactly those which eliminate quantifiers and that class II theories are exactly those which satisfy relative quantifier elimination. Dense linear Order without first or last element is a class I theory; dense linear order with first and no last element, with last and no first element and with both first and last elements are all class I I theories. All nuclear theories (el. Schmerl [18]) are either class I or class II, hence all No-categorical trees are either class I or class II. Shelah has given an example of an axiomatizable (but not finitely axiomatizable) class

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328 G. Weaver

III theory (cf. Clark and Krauss [7]). It is shown below that for any prime p, the theory of infinite Abelian groups all of whose non-zero elements are of order p, when formulated in relational constants, is class III.

Using •ideas of Paillet [12] a subclass of the class II theories is characterized. These theories, called pseudo class I theories, are characterized by the property that for some n, all models of T are n-elementary. This class is shown to contain all nuclear class II theories. Further, by the addition of only finitely many new predicates, any pseudo class I theory can be extended to a class I theory, preserving the number of models and the number of complete n-types with parameters (Lemma 4.1).

Clark and Krauss [7] contains an extended discussion of the algebraic interest in this classification. It is also possible that the classification may provide a way of systematically attacking various questions about No-categorical theories. For example, consider the question of whether or not there are finitely axiomatizable theories which are categorical in all infinite powers. While a negative answer to this question was provided by Zil'ber [23], many partial results obtained in the 70's are naturally seen as applying to one or another of the above classes of theories. For example, from results of Mazoyer [11] it follows that no finitely axiomatizable class I theory is ~o-stable or categorical in any uncountable power. Further, it follows from results of Tait that any class I, o~-stable theory has only finitely many models in each infinite cardinality (cf. Baldwin [1]). Both of these results are easily extended to pseudo class I theories. In Schmerl [18] it is shown that all nuclear theories are finitely axiomatizable. Thus, since all such theories are essentially class I, no nuclear theory is 0~-stable or categorical in any uncountable power. This result also follows from the observation of Belegradek [3] that no nuclear theory has a model with an infinite set of total indescernibles. Belegradek also observed that atomless boolean algebra is a finitely axiomatizable class I theory which is not nuclear.

The above discussion suggests several questions which to the author's knowledge are open:

(1) are there finitely axiomatizable class III theories? (2) are there class II theories which are not pseudo class? (3) are there finitely axiomatizable class II theories which are not nuclear? and (4) are there finitely axiomatizable class II theories which are not pseudo class I?

Notice that a negative answer to (1) together with negative answers to any of the others would imply Zil'ber's theorem.

1. Let K be a finite set of non-logical constants. We assume for the moment that K contains no functional constants. While this restriction is essential to some of the results below, it is not essential to the classification since the set of models of an No-categorical theory is uniformly locally finite (cf.

Classifying No-categorical theories 329

Clark and Krauss [7], p. 261). Let L~ denote the first order l_anguage (with equality) over the set K. Interpretations for L~ are ordered pairs d = (A,f~) where A is a non-empty set (the domain of d ) and f d is a function defined on K in the usual way. TK denotes the proper class of interpretations for L K. d - ~ indicates that d and ~ are elementarily equivalent; d ~ ~ indicates that d and ~ are isomorphic;f: d . ~ Y) t h a t f is an isomorphism from d to ~; d _~ ~ that d is a submodel of ~ ; and d < ~) that d is an elementary submodel of ~.

For d a Tr and B ~ A, d [B] denotes that submodel of d determined by B. A[B] denotes the domain of d[B] . Recall, if K contains no individual constants, then A [B] = B; if B = A and K contains no individual constants, d [ B ] is not an interpretation for L K. For 49 a formula in Lr in n free variables d(49) denotes the set of ordered n-tuples of elements in A which satisfy 49 in d . F o r f any function, D(f) denotes the domain off , R(f) denotes the range off . When no confusion results, we let ~ denote a sequence of members of A and f (d) = f ( a 0 . . . f ( a , ) when ~ = a a ... a,. S K denotes the set of sentences in L K. For each n a co, SK [n] denotes the set of sentences in Lr of quantifier rank <~ n. d - , N indicates that d and N agree on all sentences in SK[n ]. When d - , ¢3 we say that d and ¢2 are n-elementarily equivalent, d ~ 49 indicates that d is a model of 49; S ~ 49 indicates 49 is a logical consequence of S; and d ~ S indicates that d is a model of S. For T a theory and e an infinite cardinal, # r(C~) denotes the number of models of T of cardinality e.

In the following, we assume some fixed enumeration of the formulas in LK. For S any non-empty finite set of formulas & S denotes (491 & .-. & 49,) and V S denotes (491 v ... v 49,) where 49i occurs before 491+1 in the enumeration. For n >~ 0, d e A " and leo~ we define T ~ a as follows: (1) 7 ' °d is the conjunction of all atomic sentences in free variables Xl, . . . , x, which are satisfied by d in d and the negations of all such formulas which are not satisfied by ~ in d ; and ~P~ld is the conjunction of &{3x,+l~P~caa: aeA} and Vx,+l v {T~caa: agA}. When n = 0, we let A = a. It is easily verified for each I that there are only finitely many formulas of the form T~ ~. kg~ d is a formula in L~ with n distinct free variables and l distinct bound variables. T~cd is the finitary analogue of Scott formulas for L,ol,,, (cf. Chang [5], p. 43). Notice that ~g~ A e S K [/]. Finally, it is easily verified for all d, a'eA", that 7s~ d and 7-'~d' are different iff d ( T ~ ) ~ d ( ~ t ~ ') = A.

For l, n e ~o, where 1 <<. n ~ l, A"/d , I denotes that partition of A" induced or generated by {kg~-"a: a e A"}" i.e., A"/d, 1 = {d(7~7"a): a e A"}, T ~ " a is called a generator of A"/d, 1. A"/d , 1 =- B"/Y3, I indicates that these partitions have the same generators. Let P(A"/s¢, k) denote that sentence which says that the generators of A"/d, k partition A" (cf. Weaver [21]). P(A"/d, k) is a sentence in S~[k]. Let P(A/~, k) = &{P(A"/s¢, k): 1 ~ n ~< k}. P(A/~, k)eSr~[k]. The relation --,, is defined as follows: d ~ o N iff d [ A ] _ -N[A] ; and ~¢ ,-~,+1N iff (i) for all aeA, there is beB s.t. ( d a ) ~ , ( N b )

and (ii) for all b e B, there is a e A s.t. (ACa)~,(Nb). The following are easily established.

330 G. Weaver

(1) (2) (3) (4) (5)

COROLLARY 1.1. d ~~N.

COROLLARY 1.2.

LEMMA 1.1 (Weaver [21]). For all n, Is o3, ~ e A% 3 e B ~ if 1 <. n <~ I, then

THEOREM 1.1 (Weaver [21]). For all leo3, the following are equivalent:

d ..~tS; d - , ~ ; for all n, 1 <~ n <<. l, A~/~¢, 1 - B"/~, l;

~ P(A/~¢, l); and ~ ~ A .

(Ehrenfeucht [9], Fra'fss6 [10]). d - 8 iff for all l,

I f d - N , then for all n, l, k e o3 where 1 <<. n <<. l <~ k, A ~ t d , l = A " l d , k iff B"/~ , l = B~/~, k.

Given d e Tr, P~: co ~ o 3 + 1 is defined as follows: (1) P~(0) = 0; (2) P~(n + 1) = I provided I is the least member of co s.t. for all k >t l, A ~ + l / d , l = A " + l / d , k, when such an 1 exists; otherwise, P~(n) = o3. P ~ is called the partition function for d . Notice that P~(n) = I iff for all deA" the type of d in d is principal and isolated by ~ - z a . Thus, P~(n) e co if all n-types of elements of A are principal iff there are only finitely many such types. It is easily verified, that if d - ~ , then P~ = P~.

For T a complete satisfiable theory in L K, let Pr: o3-,c0+ 1 be s.t. PT = P.~, for d ~ T. P r is called the partition function for T. It follows from results of Rosenstein [16], that for each n >/1, there is K and T a complete satisfiable theory in L r s.t. Pr(n)eo3, but P r ( n + l ) = co.

Given d , ~ e T r and f a map, f is a partial isomorphism between d and iff D(f ) is a f ini tesubset of A, R ( f ) ~_ B a n d f i s an isomorphism between

~¢[O(f ) ] and 8 [ R ( f ) ] ; f is an elementary partial isomorphism between ~[ and ~ iff there is n~> 1 s.t. D ( f ) = {a 1 . . . . . a,} ~ A, R ( f ) ~ _ B and ( d a 1 ... a ~ ) - ( ~ f ( a t ) . . . f ( a ~ ) ) ; for neo3, f is an n-elementary partial isomorphism between d and ~ iff there is m >i 1 s.t. D(f) = {a I . . . . , a,~} ~_ A, g ( f ) ~_ B and ( d a 1 ... am) - ~ ( ~ f ( a l ) ... (aN)), f is a partial (elementary partial) (n-elementary partial ) automorphism on d i f f f is a partial (elementary partial) (n-elementary partial) isomorphism between d and d . It is easily verified that the partial isomorphisms (automorphisms) are exactly the 0-elementary partial isomorphisms (automorphisms).

For ~¢, ~ countably infinite members of T r, P I ( d , ~ ) denotes the set o f partial isomorphisms between d and ~ . Let Z ~ P I ( ~ , 8) , Z has the back and forth property iff (1) for a l l f e Z, a e A, there is g ~ Z s . t . f _~ g, and a e D(g); and (2) for a l l f e 2;, b e B there is g e Z s . t . f _ g and b e R(g). It is easily verified that if 2; _~ P I ( d , ~), Z ~ A and Z has the back and forth property, then for a l l f e Z, there is g: d ~ ~ s . t . f _~ O (cf. Barwise [2]). Let E P I ( d , ~) denote the set of elementary partial isomorphisms from d to N; let EPI,(s¢, 8 ) denote the set of n-elementary partial isomorph.isms from d to N; let PA(s¢)

.Classifying No-categorical theories 331

denote the set of partial automorphisms on d , E P A ( d ) denote the set of elementary partial automorphisms on d and E P A , ( d ) denote the set of n-elementary partial automorphisms on d . For T a complete theory and d , countably infinite models, of T, S (PT) (d , &) = {f: for all n, if card D(f) = n, and er(n) e co, then f e E P I e ~ , ~ - , ( d , ~)}.

THEOREM 1.3. For T a complete satisfiable theory

(1) if for all n ~ o3, PT(n) ~ o3, then for all ~ , ~ countably infinite models of T, S ( P r ) ( d , ~) ~ A and has the back and forth property;

(2) Pr(n) ~o3, for all n eo3, iff T is No-Categorical.

PROof. Let T be a complete satisfiable theory.

(1) Suppose that for all n z o3, Pr(n)e o3. Let d , ~ be countably infinite models of T. Since T is complete, d = g . Thus, d -¢e~.¢1)-1)+1 ~- Therefore, for all a e A, there is b e B s.t. ( d a ) - vT<I)- 1 (gb). Thus, ~ ( P r ) ( d , ~) # A. L e t f e X ( P r ) ( d , ~) . There is n I> 1 s.t. D ( f ) = {a l , . . . , a,}. Let fi be a t ... a,. Then ( , ~ ) - e-~t,~-,(~f(d)). By Lemma 1.1, ~ ~ ~e~Tt")-"a[f(a)]. ~Tt"~-"a is a generator of A " / d , PT(n), Pr(n)~< P r ( n + 1) and A " / d , P r ( n ) = A " / d , PT(n+I) . Thus, T ~ V x 1 ... x , (~r~")-"a = ~¢r~"+i)-"a). Therefore, I = 7'P~'t"+l)-"arf t a v l ~ , L~ t J_J. Hence, by Lemma 1.1, (da )~eT<,+ i ) - i (~f(a)). Thus, for all a e A, there is b e B s.t. (daa)=p~.~,+i)_~,+l)(~f(a)b); and vice versa. Thus 27(Pr)(d, ~) has the back and forth property.

(2) Suppose Pr(n)e co, for all n. By (1) 2~(Pr)(d, ~) # A and has the back and forth property. Thus, d ~ ' and T is No-categorical. Suppose T is No-categorical. By the Ryll-Nardzewski theorem, T has only finitely many n-types for each n; and each such n-type is principal. Let a e A", and let ~b isolated the type of a in d . Let t(a) denote the number of distinct bound variables in q~. ~ a isolates the type of a in ~¢. Let t = max{t(a): a e A}. For all l >>- t + n, A " / d , l = A " / d , t + n; thus Pr(n) ~< t + n.

Theorem 1.3 tells us what partial isomorphisms are available for con- structing isomorphisms between the countable models of T; further for d a countably infinite model of T and f a partial automorphism on d , if cardD(f) = n and f is a (Pr(n)-n) elementary partial automorphism on ~¢, there is ,q an automorphism on d s.t. f _ g.

When K contains functional constants, it is in general not possible to define 7'~ta. However, when T is No-categorical, the models of T are uniformly locally finite (Clark and Krauss [7], p. 261) and under these conditions ~ , a can be defined (cf. Lemma 1.11, Clark and Krauss [7], p. 260). For this reason, the assumption that K does not contain functional contains is not essential for the results of the next section.

2. Henceforth, unless stated otherwise, we restrict attention to complete, No-categorical theories with infinite models. Let C(I) denote the set of class I theories; C(II), the set of class II theories; and C(III), the set of class III

332 o. Weaver

theories. It is easily verified that each theory is in exactly one of these classes. The following lemma plays a central role below.

[,EMMA 2.1. For T a theory, and d a countable model of T

(1) Te C(I) /ff PA(s~) has the back and forth property; (2) Te C(II) iff PA(~) does not have the back and forth property and there is

n >>, 1 s.t. EPAn(d ) has the back and forth property; (3) Te C(III) iff for all n >i O, EPA,(d) does not have the back and forth

property.

PROOF. Let T be a theory, d a countable model of T.

(1) Suppose TeC(I) . Then, PT(n)= n for all neon. By Lemma 1.2, S,(PT)(d, d ) has the back and forth property. We claim that S,(PT)(d, d ) = PA(d) . To see this note that EPII,~.~,)_,(d, d ) = EPIo(d, d ) = PI (d , d ) = PA(d). Suppose that PA(d) has the back and forth property. Recall that

PA(d) ~ A. Thus, for all m/> 1 all d , /~A m, i f ( r id) - 0 (rib), (r id) - (rib-) and (r id) = (db-'). Thus, for all a ~ A", d ~ V x 1 ... x m ~o a -- ~'~ a for all t ~> 0. Thus Am~d, m = Am~d, l for all l>~ m. Thus, Pal(m)= m. By definition, P r = P~¢. Therefore, T is a class I theory.

(2) Suppose TeC(II). Then, there are k>/ 1 and l > 0 s.t. for all n~>l , Pr(n) = n+k. We claim that EPAk(d ) has the back and forth property. By Lemma 1.2, S(Pr) (~ ' , sO) has the back and forth property. By (1), PA(d) does not have the back and forth property. It suffices to show that z~(PT)(.~I , ~1 )= EPAk(d ). For all t, PT(n)--t <~ k. Thus, EPAk(d )

Z(PT)(d, d ) . By Theorem 1.3, Z(PT)(d, d ) ~_ EPAk(d ). Suppose PA(d) does not have the back and forth property, but there is m >~ 1 s.t. EPA,~(d) has the back and forth property. Let k be the least such member of ¢0. By reasoning as in (1), PT(n) <<, n+k, for all n. For all n there is l(n) s.t. PT(n) = n+ l(n). Further , l(n) ~ l(m) for all m, n < m. By our choice of k, there is t s.t. l(t) = k; hence, for all n > t, l(t) = l(n). Thus, for all n ~> t, Pr(n) = n+l(n) = n+l(t) = n + k; and Te C(II).

(3) Suppose Te C(III). Then T¢ C(I) and T¢ C(II). Thus, by (1) and (2), for all n, EPA,(d) does not have the back and forth property. Suppose that EPA,(d) does not have the back and forth property, for all n. Suppose T¢ C(III). Then, there is k >t0 s.t. for all n, PT(n) <<. n+k. Suppose k = 0. Then, Pr(n)= n. Thus, by (1), EPAo(d ) has the baci~ and-for t la property. Suppose k > 0. We may assume that k is the Smallest member of 0~ s.t. Pr(n) ~< n + k. Thus, by our choice of k, there is t s.t. l(t) = k and for all n >I t, Pr(n) = n + k . Thus, TeC(II) and by (2), EPAk(d ) has the back and forth property. Thus, in either case, there is n s.t. EPA~(d)has the back and forth property, contradiction.

This classification is related to a not ion of the structural complexity of


countable models. This not ion is arrived at by considering the conditions under which isomorphisms between finite submodels can be extended to automorphisms. In the case of class I theories, any isomorphisms can be extended (i.e., if ( d a ) "~o (rib-J, then (r id) ~ (d/~)). Here, all that is relevant is how the objects in the domains of the submodels are related among themselves. The countable models of class II theories are more complex. These theories have isomorphisms between finite submodels which cannot be extended to automorphisms. Whether or not a given isomorphism can be extended depends on how the submodels sit insidr" the model. Thus, in contrast to class I theories, it is relevant how the objects in the domains of the submodels are related to other objects in the model. For example, if for all n > 0, P2.(n)= n + k , then if ( r id) ~k (~¢b-'), then ( d a ) - (d/~). The value of k indicates the complexity of the relationships to be considered. For example, if k = 1, what is relevant is relationships between elements of the domains of the submodels and single points in the model; but, if k = 2, what is relevant is relationship between the elements of the domains and pairs of points in the model. For class II theories, the complexity of the relevant environment is eventually independent of the cardinality of the sets generating the submodels (i.e., there is k and l s.t. for all n >>. l Pr(n) = n+ k). This is not true for class III theories.

Let K = {R), where R is a binary relational constant. Let T be the theory of dense linear order without first or last element. Let d be any countable model of T, it is easily shown that P A ( d ) has the back and forth property (cf. Barwise [2]). Let T 1 be the theory of dense linear order with first but no last element. Let d be any countable model of T 1. Notice that PA(~¢) does not have the back and forth property. However, it is easily verified that, E P A I ( d ) has the back and forth property. Thus, T ~ is a class II theory. Examples of class III theories appear to be hard to come by. Further, the author knows of none which are finitely axiomatizable.

3. There are alternative characterizations of the above classification. Here we consider two of them. For n ~ o3, d is n-homogeneous iff for all m/> 1, a,/~eA", if ( r i d ) = , ( r i b ) , then ( d ~ ) ~ (db-). Thus, d is n-homogeneous provided E P A , ( d ) has the back and forth property. 0-homogeneous models are exactly the ul t rahomogeneous models (cf. Rose and Woodrow [14]). Notice that when d is n-homogeneous, d is m-homogeneous for all m ~> n. Let d be any countable homogeneous model, h ( d ) denotes the least n s.t. d is n-homogeneous; otherwise h ( d ) = co. Following Clark and Krauss [7], h ( d ) is called the homogeneity rank o f d . The following lemma is an immediate consequence of Lemma 2.1.

LEMMA. 3.1. For all T

(1) Te C(I) iff for all d a countable model of T, h ( d ) = 0; (2) T~ C(II) / f f there is n >~ 1 s.t. for all d a countable model of T, h ( d ) = n; (3) Te C(III) iff for all d a countable model of T, h ( d ) = o3;

334 G. Weaver

For ~b any formula in L r let V(~b) denote the variables occuring free in q~. For each n >/ 1, let Fx[n ] = {tk: V(~b)_ {x~, . . . , x,}}. For each n, m let Fx[n, m] = {~b: ~b e Fr[n ] and ~b contains at most m distinct bound variables}. Let T be any theory (not necessarily complete or No-categorical), and let m >>. O, T eliminates all but m quantifiers provided for all n all q~ e F r In], there is ~keFr[n ,m ] s.t. T ~ ~'x 1 ... x , ( ~ - ~b). Let Q(T) denote the least m s.t. T eliminates all but m quantifiers when such an m exists; otherwise Q(T) = co. T eliminates almost all quantifiers provided there is n >t 1 s.t. Q ( T ) = n. T eliminates quantifiers provided Q ( T ) = 0.

LEMMA 3.2. For all T,

(1) T~C(!) /ff Q(T) = 0 ; (2) T~COI) /ff Q(T) = n, n >>. 1; and (3) T¢ C(III) /ff Q(T) = to.

4. Elimination of almost all quantifiers is but one natural generalization of quantifier elimination. There is another generalization which is of interest in the present context. For S a set of formulas, S is finitely generated provided there is S' a finite subset of S s.t. for all q~ e S, either q~ e S' or there is ~k e S' s.t. ~b is the result of replacing some (perhaps all) of the free occurrences of variables in ~ by other variables provided the "new" occurrences of these variables are free in ~b. S' is said to generate S. For T any theory, T eliminates all quantifiers but those in S iff for all n, q~ e F x In], there is ~, a truth functional combination of members of S s.t. V(q~) = V(~k) and T ~ Vx 1 ... x,(q~ = ~k). When T eliminates all quantifiers but those in S and S is finitely generated we say that S is a set of basic formulas for T. Talmost eliminates quantifiers iff there is S a set of basic formulas for T and for all S a set basic formulas for T at least one member of S contains a bound variable. It is easily verified that if T almost eliminates quantifiers, Q(T) < to. Hence if T almost eliminates quantifiers and is No-categorical, then Te C(II).

For Ta complete, No-categorical theory with infinite models, T is a pseudo class I theory iff T almost eliminates quantifiers. Let PC(I) denote the set of pseudo class I theories. By the above remarks PC(I) G C(II). Let K = {R} where R is a binary relational constant; let T 1 be the theory of dense linear order with first but no last element. T 1 e PC(I) (cf. Chang and Keisler [6], pp. 58-59).

Let T be any theory which almost eliminates quantifiers and let S be a set of basic formulas for T where {~b 1 . . . . , ~b,} generates S. For each i, 1 ~< i ~< n, let 2i = x~l ... x~,. denote the variables Occurring free in the, let R~ be an m-ary relational constant not in K and let K * = K w {R~: 1 ~<i~< n}. Let A = {V~,(~b i =_ R,(~,)): 1 ~< i ~< n} and T* = T u A. Interpretations in Tr* are denoted zz¢* = ( d , Rff*, . . . , Rff*) when ~1 e T r and R~* is the extension of Ri in ~¢*. For d e T K, d ( A ) = ( d , d(q~0, . . . , ~'(q~,)). ~¢(A) ~ A; and if d ~ T, d ( A ) ~ T*. Notice that, if ~ ¢ * ~ T*, ~¢* = d (A) . The following is easily verified.


LEMMA 4.1. For all theories T (not necessarily complete or No-categorical), i f T almost eliminates quantifiers, then

(1) T* eliminates quantifiers; (2) for all W , ~ , if d h T then d _~ ~ iff d ( A ) ~ ~(A); (3) T is complete iff T* is complete; (4) T is finitely axiomatizable iff T* if finitely axiomatizable; (5) for all d ~ T, B ~_ A , the number of n-types realized in d B equals the

number o f n-types realized in d(A)B; (6) for all ~ , all m >>. 1, a, 6cA m, ( d a ) - (d6 ) /ff (d(A)a) ~- (d(A)[~).

From Lemma 4.1, we immediately obtain:

THEOREM 4.1. For T a theory, if Te PC(I), then

(1) T*aC(I) ; (2) T is fn i t e ly axiomatizable iff T* is; (3) for all cardinals ~, ~ r ( ~ ) = ~r.(~); (4) for all cardinals a, T is s-stable iff T* is ~t-stable.

Let d be any interpretation, n ~> 1, following Paillet [13] we say that d is n-elementary iff for all f e P A ( d ) if for all B ~ A s.t. card B = n, f r B ~ EPA(.~), t h e n f e E P A ( d ) . Notice that when T is a class I theory, then for all n s.t. every predicate constant in K is of degree ~< n, all countable models of T are n-elementary. Further, if h ( d ) = 0, and n is as above, d is n-elementary. We show below that when T is a class II theory, Tis a pseudo class I theory provided there is n s.t. all countable models of Tare n-elementary.

X I , - . . , X t l - 1 , X r t For ~ a formula, xl , . . . , x,, Yl . . . . , y, variables, (~)[y, ..... y,-1,y,] denotes the result of simultaneously replacing every occurrence of x i in ~ by Yi.

PROPOSITION 4.1. For all d , n >~ 1, if n is greater than or equal to the degree o f all predicate constants in K, then the following are equivalent

(1) d is n-elementary (2) for all re>In, d e A m, geA m, a, b eA , if ( d d ) = ( d ~ ) and for all

i x < ... <i,_~ <~ m , ( d a q ... ai,_,a) = ( d b i , . . . b i , _ , b ) , then (Jda) - dbb).

PROOF. Suppose d is n-elementary. Let m ~> n, a = a~ ... a,,, = b~ . . . . . b,,. Suppose ( d d ) - ( d ~ and that for all i~ < ... < i,_ 1 <~ m, (da~, . . . ai,_~a) - (Wbi~ . . . bi._,b). Let f : {a 1, . . . , a,,, a ) -~ {bx, . . . , b,,, b} be s.t.f(al) = b i and f ( a ) = b. Since n/> degree of all predicate constans in K, f e P A ( d ) . Let B ~_ D( f ) s.t. cardB = n. By supposition, f I B e EPA (~/). Thus, f e EPA ( ~ ) and (~'da) - (~'6b).

Suppose (2). We proceed by induction to show that for all m > n, all a , 6cA m, if for all il < ... < i, <~ m, (dai , ... ai.) =- ( d b ~ ... hi,), then (Wa) = ( d S ) .

THEOREM 4.2. For all T, if Te C(II), then Te PC(I) iff there is n >~ 1 s.t. all countable models of T are n-elementary.

336 G. Weaver

PROOF. Let TeC(I I ) , ~ T, d countable. Thus, there is m >1 1 s.t. h ( d ) = m; and for all r >i 1, 8 e A ' d ~ Vx 1 ... x,(O~a - O~a) for all l >/m. Suppose there is n >i 1 s.t. all countable models of T are n-elementary. Let A = {O~a: a e A', 1 ~< r ~< n} u {x 1 ¢ xl}. Let S~ be the set of all substitution instances of A. Sa is generated by A. We claim that T eliminates all quantifiers but those in S d. Notice that, if T has a set of basic formula which contains no quantifiers, Q(T) = 0 and T¢ C(II). Thus, if T eliminates all quantifiers but those in S~, T almost eliminates quantifiers. Let t > n, it suffices to show that if a ~ A ' , ~?~a, is equivalent to a truth functional combination of sentences in S~. Let a = a 1 ... at, since d is n-elementary, d t= VX 1 ... xt(ff~a = {~b~ail a,n[~11...~;i,]: for all i~ ... i, ~< t}). Thus, TePC(I ) .

Suppose Te PC(I). Let S be a set of basic formulas for T generated by {~b~ . . . . , q~k}" Let {x~, . . . , xt} w {V(q~,): 1 ~< i ~< k}. Let t* be s.t. t* >t t and every predicate constant in K has degree ~< t*. We claim that d is t*-elementary. We proceed by applying Proposition 4.1. Let r > t* and a = a~ ... a, e A ' , [~ = b 1 ... b, eA" and a, b e A . Suppose (~¢8) = (~'/~) and (~¢a~ ... ai,. ,a) - (~¢bi~ ... bi, ~b) for all i 1 < ... < it,-1 <~ r. Then, (~¢(A)a) - (~¢(A)/~) and (d(A)a~, ... a~-._,a) = (d (A )b~ ... b,,._~b). Thus, (d(A)~a)

=-=° (~¢~(A)[Jb)'(dbb). .by our choice of t*; and (~¢(A)aa)=-(~¢(A)~h). Thus, (o~¢aa)

Thus, if T is pseudo class I, and ~¢ is a countable model of T, there is n s.t. is n-elementary. Hence, whether or not an isomorphism between ~ and cg,

finite submodels of ~¢, can be extended to an automorphism on M depends upon how those submodels of ~ and ~ generated by n-element sets sit inside of ~ ' .

Given ~¢, n >i 1, A is n-nuclear iff for all m >~ n, a e A r~, a e A , there is i~ < ... < i~ ~< m s.t. for all /~ e A m, b e A, if (~¢a) = (~¢/~) and (~¢all ... a~a) = (~¢bi~ .. . b~,b), then (~¢aa)- (~¢/~b). It is easily verified that if ~¢ is n-nuclear, ~¢ is m-nuclear for all m ~> n. ~¢ is nuclear iff there is n ~> 1 s.t. ~¢ is n-nuclear. It is immediate from Proposition 4.1 that if ~ ' is nuclear, there is n s.t. ~¢ is n-elementary. The converse, however, does not hold (see below).

THEOREM 4.3. (Schmerl 1-18]). I f all countable models of T are nuclear, T is finitely axiomatizable.

Using the above ideas, Theorem 4.3 can be strengthened slightly: if d is a countable model of T and ~ ' is n-nuclear and no predicate constant in K has degree > n, f f~t2"÷~A finitely axiomatizes T.

COROLLARY 4.1. I f all countable models of T are nuclear, Te C(I )u PC(I).

While nuclear models are n-elementary for some n. There are n-elementary models which are not nuclear. By Theorems 4.2 and 4.3, it suffices to find a pseudo class I theory which is not finitely axiomatizable. Let K -- {R} where R is a binary relational constant. Let d be a countably infinite interpretation


for L r where there is unique a in A and B _ A s.t. (1) (a, a ) e f t ( R ) ; (2) if (c, b) ef~c(R); then c = a; (3) b e B iff (a, b) ~f~,(R); and (4) both B and A - B are infinite. Note that B = d ( 3 x R x y ) . Let T be the complete theory of d . It is easily verified that (1) T is No-categorical; (2) T is not finitely axiomatizable; (3) for all n > 0, Pr(n) = n + 1; and (4) all countable models of T are 1-elementary. Thus, T is a pseudo class I theory.

5. In this section we show that there are No-many class III theories. Let K = {0, + , - } where + is a predicate constant of degree 3, - is a predicate constant of degree 2 and 0 is an individual constant. For each prime number p, the theory Tp can be formulated in S~; where Tp is the theory of infinite Abelian groups all of whose non-zero elements are of order p. For each p, Tp is categorical in all infinite powers (cf. Bell and Slomson [4], p. 180). For d a model of Tp, + ~ will d e n o t e f ~ ( + ) ; - ~ , f d ( - ) ; and 0~,f~(0). When no confusion results, the subscripted d will be deleted.

Let K* = { + *, - * , 0} where + * is a binary functional constant, and - * is a unary functional constant. For each prime p the theory of Abelian groups all of whose non-zero elements are of order p can be formulated as a universal theory in LK,. Let Tp* denote this theory. Let T~ denote the theory of the infinite Abelian groups all of whose non-zero elements are of order p. Tp" and Tp are synonymous in the sense of DeBouvere [8]. Given d ~ Tp, let d * be the corresponding model of Tp '°.

PROPOSITION 5.1. For all p, T~' eliminates quantifiers.

PROOF. Notice that all models of Tp '~ are infinite, any union of a chain of models of T~ is a model of Tp and Tp ~ is a-categorical for all infinite a. Thus, by Lindstr6m's theorem (cf. Chang and Keisler [6], p. 114) Tp ~ is model complete. Further, T~ is the model completion of Tp*. To see this note the following: (1) every model of T~ is a model of T*; (2) every model of Tp* is a submodel of a model of T~; and (3) if d * is a model of Tp °' and ~ * , cg, are models of Tp* where d * _ ~ * and d * _ c~,, then ~* and cg, are models of T~'°; and since ~** and oK** are models of T~ w D(d*) , ~** = cg~,. By a theorem of Robinson (cf. Sacks [17], p. 67), since Tp* is a universal theory and Tp ~° is the model completion of T*, Tp eliminates quantifiers.

We proceed below by computing the partition function for T 2 and showing that for each p, Prz(n) <~ Prp(n) for all neo. We define a function g: 0 3 ~ o s.t..

Pr2 (g(n)) = g(n) + n.

Hence, for all l

l+g( l+ 1) < PT2(g(l+ 1))

and T 2 is class III. g is defined as follows: g(0) = 3, and g(n+ 1) = 2(g(n)-1) . The proof that PT2(g(n)) = g(n)+n comes in two parts. First, it is shown that

338 a. Weaver

any n-elementary partial au tomorphism between sequences of length g(n) can be extended to an automorphism (Theorem 5.1); and secondly, it is shown that there is an (n-1)-e lementary partial au tomorphism between sequences of length g(n) which cannot be so extended (Corollary 5.1). The proof of Theorem 5.1 proceeds by exhibiting an algebraic condition on partial automorphisms which implies that they can be extended to automorphisms (Lemma 5.2), and by finding a formula which is satisfied just in case that condition holds• The proof of Corollary 5.1 proceeds by finding an algebraic condition on partial automorphisms which is equivalent to their being n-elementary partial automorphisms (Theorem 5.2)•

LEMMA 5.1 For all n, I ~ g(n), there is c~(xl, . . . , x~)eFK[l, n] s.t. for all o f ~ T 2, all a 1 ... algA.

l

of ~ qS(xl,. . . , x t ) [ a l , . . . , at] iff ~ a t = 0. i = l

PROOF. Proceed by induction on n. Let n = 0. Then, g(n) = 3. Let I be s.t. 1 ~< l ~< 3. Suppose l = 3, tk(x 1, x 2, x3) is + (xl, x2, xa). If I = 2, ~(x 1, x2) is + (xl, x2, 0); and if l = 1, tk(xl) is x I = 0. Suppose that the above holds for t. To show that it holds for t + l . L e t / be s.t. 1 ~ < l ~ g ( t + l ) . Let l = g ( t + l ) = 2 (g ( t ) - l ) . By hypothesis, there is (p(x l , . . . , xg to) s.t. for all

a 1 , • . . , ago)cA,

off ~ dp(xl, . . . , xg<t)) [a 1 . . . . . a g J

Let ~ ( x g ~ t ) + l , . . . , x2<o<o)) be

g ( t )

iff ~ a i = 0. t = l

( ¢ ( x l x lr-, ..... -,<,, 1 " " • ' g(t)J] L~Cg(t) + 1 , . . . , 2 g ( t ) - I "

Let ~b(xl, . . . . xgt0-1,xg<0+ 1, . .-, x<2g~0-1) be

• • • , g ( t ) Y ] L y n ÷ 1 _1 " " " ' 2 g ( t ) l ] L y n + 1 ..1! •

This formula contains t + 1 distinct bound variables and 2(,q(t)- 1) = g( t+ 1) free variables. Let l < g ( t + l ) . Let qS(x I . . . . ,xt) be (~p(Xl,. . . ,xg<o_t, xgtt)+l, . . . , x2a,)-O)[~',+..~..'6"'x~"~-~]. It is easily verified for all a 1 . . . . . areA, that.

iff

o f ok(x1, . . . , x )[al . . . . , a d

l

a ~ = 0 . i = 1

L E M ~ 5.2. For o f ~ T a s.t. cardof = N o, l /> 1, B = { b l , . . . , bt} ~-- A, t t

f : B ~ A, if f is 1 - 1 and for all t s.t. 1 <. t <~ l, ~ bt~ = 0 iff Y' f (b~) = O, then j = l j - 1

there is h s.t. f ~_ h and h is an automorphism on of .


PROOF. It follows from Proposi t ion 5.1, that T* admits the elimination of quantifiers. Hence, T* is submodel complete. Further, since T* is No-categorical, all models of 7"2* are homogeneous.

Let ,~¢~ T 2 where card ,~¢ = N O . Given l >t 1, B = {b~ . . . . . bt} ___ A and ± ,

f : B--~A, suppose that f is 1 - 1 and for all t..< l, b o = O i f f ~ f ( b o ) = 0 . j = l j = l

Then, there is f ' s.t. f : _ f ' and f ' : zC*[Bl ~ z C * [ f I - B ] ] . ~¢*[B] and ~ ¢ * [ f [ B ] ] are submodels of z¢*, and as T2* is submodel complete, (~¢*b~ . . . b , ) - (zC*f(b 0 . . . f (b , ) ) . Since z¢* is homogeneous, (zC*b~ ... b~) -~ (~¢*f (b 0 . . . f (b,)). Thus, (zCb~ .. b~) ~- (zCf (b 0 . . . f (b~)). Hence, there is h s.t. f _ h a n d h is an automorphism on z¢.

The following is immediate from Lemmas 5.1 and 5.2.

THEOREM 5.1 If ~ t ~ T 2 and card ~¢ = N 0, then for a 1 . . . a t ~ A , b 1 . . . b t e A and n s.t. n = m i n { t : 1<~9(0} , = - , ( d b 1 . . . bl), then (~a~ . . . at) ~- (egb~ .. bt).

all l >~ 1, all

i f (,.~¢a I . . . at)

PROOF. By the choice of n, l <~ g(n). Thus, if ( d a 1 ... al) - , ( r ib 1 ... bt) t t

then for all t~<l , ~ % = 0 i f f ~ b i j = 0 , by Lemma 5.1. Without loss of j=l j=l

generality, we assume that a~ # ak, when i # k. Hence, by Lemma 5.2,

( d a 1 . . . a~) ~ (sgb 1 . . . bt).

It is immediate from Theorem 5.1 that for all n /> 1, Pr2(n) <~ n + k where k = min{t: n <~ g(t)}; hence, that eTe(g(n)) <~ g(n)+n. The following theorem implies that PT2(g(n)) = g(n) + n.

THEOREM 5.2. For n >f O, t > O, 8, [~eA t and .~¢~ T 2, (.4~l) ~n (,4~) i f f 1 l

for all l<~ g(n), Y, ai r = 0 iff y ' b,j = 0. 3=1 j=l

PROOF. Let d ~ T z. Proceed by mathematical induction. Let n e T i f f l

n/> 0, for all t > 0 all 8,/~e A', ( d a ) - , ( d ~ ifffor all 1 <<, g(n), ~ a 0 = 0 iff r l = l

l

~, bij = 0. Let t arbitrarily chosen. Let 8, [~eA t. Suppose that (~¢d) -=o (~¢/~). j = l

By Lemma 5.1, for all l ~< g(0) = 3, there is ~(Xl , . . . , x~) ~FK[I, 0] s.t. for all l

ai~ . . . . . ai,, ~¢ ~ 49(xl . . . . . Xl)[al~ . . . . . ai, ] iff ~, aij = 0. Thus, by supposition, j = l

l l l t

E a0 = 0 iff E bis = 0. Suppose that for all I ~< g(0), ~ a,j = 0 iff E b0 = 0. j = l j = l j = l j = l

Then, a~ # aj iff bi # bj. It suffices to show that there is j¢: ~'[{a~ . . . . . at} ] ~- -~¢[{bl, . . . , bt}], wheref(a~) = b i. Let f(0) = 0 andf(ai) = bi. By supposition,

f is well defined, 1 - 1 and onto A[{b 1 . . . . . bt} ]. Recall that ai = - a j , iff

340 G. Weaver

a~ + aj = 0, and that ai~ + a~2 = ai3 iff a~ + a~2 + a~ = 0. Hence, a~ = - aj iff b i = - b j and a i l q - a i 2 : ai3 iff bi~q-bi2 = bi3.

Suppose that k ~ T. To show that k + 1 e T. Let t be arbitrarily chosen. Let a, b ~ A t. Suppose that (~ 'a) --k+ ~ (S~/~). The result follows from L e m m a 5.1.

l !

Suppose for all l ~< 0 ( k + 1), ~ aij = 0 iff ~ b~j = 0. Since g(k) < g ( k + l ) , by j = 1 j = l

induct ion hypothesis ( d ~ ) =1, (db-'). I t suffices to show that (a) for all a there is b s.t. ( ~ a ) - k + 1 (~¢/~b) and (b) for all b, then is a s.t. (s4aa) - k + 1 (d/~b). We consider (a); the a rgument for (b) is analogous. By induct ion hypothesis, given

l

a it suffices to find b s.t. (1) a i -~ a iff b i ¢ b; and (2) for all l < g(k), ~ aij+a j = l

I

= 0 iff ~ b~j + b = 0. We proceed by cases on a. Case (1). Suppose that a = a i j = l

for some i. Let b = b~. Case (2). Suppose that a = 0. Let b = 0. Case (3). l l

Suppose there is l < g(k) s.t. a = ~ ai~. Let b = ~2 bi~. Case (4). Suppose j = 1 j = 1

l

a ~ O, a ~ a i, for all i; and for all l < g(k), a ¢ ~ aq. Let b be s.t. b ¢ b i for j = l

l

all i; b 4 = 0 and b e ~ b~. j = l

(1) (2)

COROLLARY 5.1.

for all ne. co, Pr2(g(n))= g(n)+n ; T 2 is a class III theory.

PROOF. (2) follows from (1).

To prove (1) it suffices for each n >~ 1 to find a,/~ e A 0(") s.t. ( d a ) = (,_ 1)(~¢~, bu t ( d S ) ~ (d/~). By Theorem 5.2, it suffices to show that for all l

l l

~< g ( n - 1 ) , Z ai~ = 0 iff Z bij = 0. No te that Pr2(g(O)) = PT2(3)+0. Let j = i j = l

~ ~ T 2 where card d = N o. Let d * be as i n the p roof of Lemma 5.2. We claim that for each n > 0 there are a 1 . . . . . a0(,)_ 1 and b s.t.

o ( n ) - 1 o(n) - 1

( d a t . . . ao~,)_ 1 • a~) =--,_~ (s la 1 ... ao~,)_lb ) where b ¢ Z a~. Let a~ be i = 1 i = 1

any member of A - { 0 } ; a 2 ¢ A * [ { a l } ] and at+lCA*[{a 1 . . . . . al} ]. Thus, g(n) - 1 g ( n ) - 1

a i • 0. Let b C A * [ { a 1 . . . . . ao(n)_l} ]. Thus, b ¢ ~' a~. Further , for i = 1 i = i

o ( n ) - 1 l ~ i ( . ) - 1

all j , aj ¢ Z ai, and aj :/= b. Finally, for all l < g ( n - 1 ) , ~] a i j+ ~ a 1 i = 1 j : l i : 1

l l

- - 0 iff ~ a i j + b - - O . By the choice of b, b:~ ~ aij. Notice that j = 1 j = l

Classifyin9 No-categorical theories 341

g ( n - 1 ) ~< g ( n ) - l . Thus, for l < g ( n - 1 ) , l < 9 ( n ) - l , and ( g ( n ) - l ) - I > O. l o(n)- 1 (g(n)- 1 ) - l

Suppose E aij + E ai = 0. Then, E aij = 0, which is impossible. j = l i = 1 j = l

It remains to show that Tp is class III, for p any prime. Since, by Corollary 5.1, l+g ( l+ 1) < PT2(9(I+ 1)), it suffices to show that Pr2(#(l+ 1)) <~ PT~(9(I+I)). As above, we find a condition which implies that ( d a ) = , ( r i g ) ; and using this condition we find a, gzA °~") s.t. ( d a ) - , _ ~ (~¢g), but ( d a ) is not isomorphic to (d6) .

TrIi~OREM 5.3. For p, a prime number > 2, d ~ Tp, n e o3, t > 0 all d, [~aA ~, l l

iffor all l ~ 9(n), ~ ni~aij = 0 iff ~ nijb~j = O, where n,j e {1, . . . , p}, then (dd) j = l j = l

----n (rib).

PROOF. As in the proof of Theorem 5.2, proceed by induction. Let n e Tif f l l

neon, for all t > 0, all d,/~aA ~, if for all l <. g(n), ~ n~aij = 0 iff ~ nibij = O,

then (~¢~) = , (m¢~). J = 1 j : 1 It is easily shown that 0 ~ T. Suppose that k a T. To show that k + 1 ~ T. Let t be arbitrarily chosen. Let

I l

gt, ~ 8 A t. Suppose that for all l ~< 9 ( k + l ) , ~ n~jaij = 0 iff ~ n~sb~s = 0. As j=1 j = l

g(k) < g(k+1), by induction hypothesis, (~¢d)=k(m¢6). As in the proof of Theorem 5.2, it suffices to show that for all a there is b s.t. (zeta) - k (~¢gb) and for all b ~here is a s.t. (dda) =k (~¢gb). We consider the argument for a. Let a be arbitrarily chosen. We proceed by considering cases on a.

Case (i). Suppose a = 0. Let b = 0. Applying the induction hypothesis,

(~¢aa) - k (d/~b). Case (2). Suppose al = a. Let bi = b. Since, (~¢d) ---k (~¢~, (~¢da) =k (~'~b).

' l

Case (3). Suppose that a = - a i. Let b = - b i. Suppose that ~ ni~aij+na j = l

1 l

= 0. Then, ~ ni~ai~+n(p-1)a i = 0 = ~ nijaij+n'ai; where n' _=vn(p-1) . j=1 j=l

l l l

Thus, by supposition, ~ %b~j+n'b~= ~" n~jb~÷n(p-l)bi= ~ %b~+b j=~ j=~ j=~

l l

= 0. In the same way, if ~ ni~bi~+nb = O, ~ ni~ai~+na = 0. Hence, by j = l j = l

induction hypothesis, (~'aa) = k (zC/~b). l l

Case (4). Suppose for l < g(k) that a = ~ nifiij. Let b = ~ n~bi. j = a j=~

l ' l ' l

Let l' < g(k), Suppose that ~ n;~ai~+na = 0. Then, ~ n;~ai~+ ~ nn,~ai~ = O. j = l j = l j = l

342 G. Weaver

I' t

Let n~ - p n n O. Then, ~ n~a'ij+ ~, n~a,j = O. l+l' <~ 2(g(k) - 1) = g(k+ 1). By j = l i = ~

I ' l l '

supposition, ~ n~b~+ ~, n~b 0 = 0 = ~ n~b'o+nb. As above, the a rgument j = l ) = 1 j = l

in the other direction is analogous. Thus, by the induct ion hypothesis

(~aa) --k (d[~b). l

Case (5). Suppose that a ~ aj, a # - a j , a # ~ ni~a 0 for l < g(k); and j = l

l

a # - 0 . L e t b b e s . t . b # b i, b # - b j , b # ~ n~jbij for all l < g(k) and b # 0. J = !

t l

Let l < g(k). It suffices to show that ~ .n~fi~:+na = 0 and ~ n~jbo+nb # 0 j = l j = It

1 l

when n # p. Suppose that ~ n o % + n a = 0. Then, ~ n~j% = ( p - n ) a and j = i j = l

l l

na = ( p - 1) ~ n~sa~j. Notice that n # p - 1. Otherwise, ~ nija~ = a. Further, j = i j = l

l l

n # 1. Fo r otherwise, a = ( p - 1 ) ~ noa 0 = ~, (p-1)nijaij . Since p is prime, j = l / = 1

n # p - -n . Thus, either n < ( p - n ) or ( p - n ) < n. Suppose that n < (19-n). Then, there is k 1 > 0 and t 1 s.t. ~ - n ) = kin + t 1 when 0 ~ t 1 < n. Since p is prime, 0 < tIt. Thus, ( p - n ) a = ( k l n + t O a = k lna+~t I ta . Note that 1 < tIt.

! l l

For if t 1 = 1, 2 n,jaij = k l n a + ~a, hence; 2 n~ja~j = k 1 2 (P-1)ni~aij+a j = l j = l j = l

l l l

and a = E n,,a,~+kl E n,~a,j = ~, (k+l)n , ja , j . ( p - n ) a = ( k I tn )a+t la j = l j = l j = i

l l l l

= E nijaij" Thus, E n o a o = k I t ( p - l ) E n,:a, i+tla and t I ta= ~, n'oa O. j = l j = I t j = l j = l

Since t 1 < n, there is k 2 > 0 and t 2 s.t. 0 ~< t 2 < t 1 and n = k 2 t l + t 2. Notice that t 2 > 0. Otherwise, n = k2tIt and p = n + ( p - n ) = k 2 t l + k l n + t 1 = k2t 1 +k~ k 2 tIt + t i = (k 2 + k~ k 2 + 1)tIt. Further , t 2 # 1. Otherwise, na = k 2 t 1 a + a.

l l I

Hence, ( p - l ) 2 ni~ai~ = k2 2 n~ja~+a and a 2 Notice that taa • t J=It j = l j = l

= ~ n~'~a O. Given r suppose there are t ,+ l , . . . , t l , and k,+l . . . . , k 1 s.t. j = l

(1) 1 < t , + 1 < t , < . . . < t 1 < n < p - - n ; (2) k I tn+q = p - n , kEq + t 2 = n, ks+lts+ts+l = t~-I for s ~< r; and

(3) t~a = ~ n*a~; s <,% r + l . J j j = !

We claim that there is t,+ 2 < t,+ 1 s.t. 1 < t , + 2 < t,+ 1 and t, = k~+2t,+ 1 +t r+2. Since t,+ 1 < t,, there is k ,+l > 0 and t,+a s.t. 0 ~< t,+ 2 < t,+ 1 s.t. t, = k,+2t~+l+t,+ 2. Suppose that t,+ 2 = 0. Then t, = kr+2t~+it. Then for all

C l a s s i f y i n g N o - c a t e g o r i c a l t h e o r i e s 343

s s.t. s < r there is d~ s.t. t, = t ,+ ld ~. To see this note that t,_ 1 = k ,+l t , +t~+ 1 = (k,+lk,+2t,+l)+t~+ 1 = (k ,+lk ,+2+ 1)t,+~. Suppose t, = d, t ,+l , t~-i = k~+lt~+t~+ 1 = k~+~d~t,+~+d~+~t,+~ = (ks+ld~+d~+l)t,+ 1. Thus, t~ = d~t,+~

and n = k 2 t ~ + t 2 = k2dl t ,+l +d2t~+ 1 = (k2dl +d2)t,+ ~ and p - n = k ln +t~ = k ld t ,+ l +d l t ,+ ~ = ( k ~ d + d l ) t , + 1 where d = k 2 d l + d 2. Thus, p = (kl d + d l) t, + 1 + dt, + 1 = ((k~ d + d l) + d) t, + ~, contradiction. Finally note that

t t,+ 1 ~ 1. Suppose otherwise. Then t ,a = k l t , + l a + a , t ,a = ~, n~a~ ~ and

l l j = l l

t,+ ~ a = ~. n~'~a,~. Thus, a = ~ n~'~' ai~. By similar reasoning, t,+2 a = ~ n~a~. j = l j = l j = l

Notice that the above is independent of r. But there are only n - 2 different numbers less than n and not l, contradiction. When p - n < n, the argument is

analogous. Thus, ~, % a ~ + na v ~ O. Similar reasoning shows that j = l

l

ni:b~: + nb v~ O. Thus, by induction hypothesis (~¢~a) --k (~¢/~b). Given b we j = l

can find a and show, by a similar argument, that (~¢8a)=--k(~b). Thus, ( b3.

C O R O L L A R Y 5 . 2

(1) for all prime p, Tp is a class III theory; (2) there are infinitely many class III theories.

PROOF. Let p be any prime > 2. ~¢ ~ Tp where card d = N o. It suffices to find for each n > O, a 1 . . . . , agtn)-b-l,-..., bgtn) in A s.t. ( d a 1 ... dgt,,)! = , - 1 ( ~ ¢ b l ... bg~)) b u t ( d a 1 ... age,)) ~ ( r ib 1 ... bg~,)). Given n, let al . . . . , agt,)-I be such that a I # 0; ma 2 ~ A * [ { a l } ] for all m < p - 1 and for s ~ < g ( n ) - l , m a s e A * [ { a I . . . . , a s - l } ] for all m < p - 1 . Let b be s.t.

g(n)- 1

mbCA*E{a 1 . . . . ,ag~n)_l}] for all r e < p - 1 . Let a = ~ a i. Clearly, i = l

( d a l . . . ag~.)-la) ~ ( ~ a 1 . . . agt~)_lb ). We claim that ( d a l .. . ag¢,)-la) =- .-1(~¢al . . . % t . ) - l b ) . We proceed by applying Theorem 5.3. Notice

that when m # p for all 1 < g ( n ) - l , ' ~ n i j a i j + m b ~ O. Otherwise, j = l

( p - - m ) b e A * [ { a l , . . . , a o ~ . ) _ l } ] . It remains to show for l < # ( n ) - i that l l

nijaij+ma ~ 0 when m ~ p. Suppose that ~ ni~ai~+ma = O. Then, j = l j = l

l g(n) - 1 g(n) - 2

ni~aij+ ~ ma i = 0. Thus, there is n' s.t. n' ~< p and n'ag~,)-i = ~ njaj. j = l i = 1 j = 2

g(n) -- 2

By construction n' = p. Thus, 0 = ~ njaj. There is a least k < g ( n ) - I s.t. j = 2

n k = m ~ 0. To see this recall l < g ( n ) - 1 and n ' ~ m. Thus, for some t, ai~= a~t,)-t and n ~ n - , p . Thus, there is k < g ( n ) - l s.t. a k ~ a~j for all

3 - Studia Logica 4/88

344 G. Weaver

j <<. I. nka k = ma k v 6 0 . Thus, there is k' < g ( n ) - I s.t. k' # k and nk, a k, ~ O. Let k* be the largest index s.t. nk.ak, 5~ O. Then, nk.ak, e A * [ { a 1, . . . , ak.-1}].

Finally, we note that the class of a theory depends essentially on the language used to formulate that theory. In particular, by Proposition 5.1 the theory of infinite Abelian groups all of whose non-zero elements are of order p is a class I theory when formulated in a language with functional constants, but class III when formulated in a language with only predicate constants. Thus, whiie Tp seems to provide the first known examples of "algebraically interesting" class III theories (cf. Clark and Krauss [7], p. 256), whether or not there are such theories formulated with functional constants appears to be an open question.

References

[1] J.T. BALDWIN, Review of Mazoyer [11], Mathematical Reviews 52 ~ 2864 (1976). [2] J. B~wtsE, Back and forth through infinitary logic, Studies in Model Theory, M. Morley (ed.),

The Mathematical Association of America, pp. 5-34. [3] O.Y. BEt~G~DEK, Review of Schmerl [18], Mathematical Reviews 55 1382 (1978). [4] J. L. BELL and A. B. St~MSON, Models and Ultraproducts, North-Holland, Amsterdam,

1969. [5] C.C. CHANG, Some remarks on the model theory of infinitary lanoua#es, The Syntax and

Semantics of lnfinitary Logic, J. Barwise (ed.) Springer-Verlag, Berlin, 1968, pp. 36-63. [6] C.C. CHANG and H. J. KEISLER, Model Theory, Amsterdam, 1973. [7] D. M. CL~K and P. H. KPa~uss, Relatively homogeneous structures, Logic Colloquium 76,

R. O. Grandy and J. M. E. Hyland (eds.), North-Holland, Amsterdam, 1977, pp. 255-282. [8] K. DE Bouc le , Synonymous theories, The Theory of Models, J. Addison, L. Henkin,

A. Tarski (eds.), North Holland, Amsterdam, 1965, pp. 402-406. [9] A. EHR~NFEUCrtT, An application of#ames to the completeness problem for formalized theories,

Fundamenta Mathematica 49 (1956), pp. 129-141. [10] R. FR~SS¢., Sur quelques classification des relations basics sur des isomorphisms restraintes,

Publications Scientifiques de l'Unlversit~ d/iiga, Set. A2 (1955), pp. 15-60, 273-295. [11] J. MAZO~R, Sur les theories cat~#oriques finiment axiomatisables, C.R. Acad. ScL Paris S~r.

A-B 281 (1975), pp. 403-406. [12] J. P~dLLET, Une ~tude sur des structures ayant une properietb de seuil pour les automorphisms

el~mentaries, C.R. Acad. Scl. Paris S~r. A-B 283 (1976), pp. 225-228. [13] J. PAILLET, Structures with a property of threshold for elementary automorphisms, Journal of

Symbolic Logic 42 (1977), pp. 463-464. [14] B. Rose and R.E. WOODROW, Ultrahomooeneous structures, Notices of the American

Mathematical Society 25 (1978), A26. [15] B. RosE, and R. E. WOODROW, Ultrahomooeneous structures, Zeitschrift flit Matematische

Logik und Grundlagen der Mathematik 27 (1981), pp. 23-30. [16] J. C. ROSENSTEIN, Theories which are not No-categorical, Proceedings of the Summer School

in Logic, Leeds, M. H. Lrb (ed.), Springer-Verlag, Berlin, 1968, pp. 273-278. [17] G.E. SACKS, Saturated Model Theory, W. A. Benjamin, Reading, 1972. [18] J. H. SCHMERL, On No-cate#oricity and the theory of trees, Fundamenta Mathematic# 94

(1977), pp. 112-128. [19] P. SCOTT, Logic with denumerably Ion#formulas and finite strings of quantifiers, The Theory of

Models, J. W. Addison, L. Henkin, A. Tarski (eds.), 1965, pp. 329-341.

Classifying No-cate#orical theories 345

1"20] W. SZMIELEW, Elementary properties of Abelian groups, Fundamenta Mathematiea 41 (1955), pp. 203-271.

[21] O. WEAVER, Finite partitions and their generators, Zeltschrifl ffir Mathematische Logik und Grundlagen der Mathematik 20 (1974), pp. 255-260.

[22] G. W~WR, Classifying No-categorical theories I, Journal of Symbolic Logic 44 (1981), pp. 682--683.

1-23] B. I. ZIL,B~, Solution of the problem of finite axiomatizabflity for theories that are categorical in all infinite powers (Russian) The Theory of Models and its Applicatlons (Russian), Kazah. Gos. Univ., Alma Ata, 1980.

[24] B. I. Zm,nER, Totally categorical theories; structural properties and non-finite axiomatizability, Model Theory of Algebra and Arithmetic, P. Pacholski, J. Wierzejewski, and A. J. Wilke (eds.), Springer-Vedag, Berlin, 1980, pp. 381-410.

DEPARTMENT OF PHILOSOPHY BRYN MAWR COLLEGE BRYN MAWR, PENNSYLVANIA 19010

Received November 20, 1987

Studia Logica XLVII, 4

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Classifying ?0-categorical theories