Ja n 19 94 UNIVERSAL THEORIES CATEGORICAL IN POWER AND κ - GENERATED MODELS

We investigate a notion called uniqueness in power κ that is akin to categoricity in power κ, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T , that if T is κ-unique for one uncountable κ, then it is κ-unique for every uncountable κ; in particular, it is categorical in powers greater than the cardinality of T . It is well known that the notion of categoricity in power exhibits certain irregularities in “small” cardinals, even when applied to such simple theories as universal Horn theories. For example, a countable universal Horn theory categorical in one uncountable power is necessarily categorical in all uncountable powers, by Morley’s theorem, but it need not be countably categorical. Tarski suggested that, for universal Horn theories T , this irregularity might be overcome by replacing the notion of categoricity in power by that of freeness in power. T is free in power κ, or κ-free, if it has a model of power κ and if all such models are free, in the general algebraic sense of the word, over the class of all models of T . T is a free theory if each of its models is free over the class of all models of T . It is trivial to check that, for κ > |T |, categoricity and freeness in power κ are the same thing. For κ ≤ |T | they are not the same thing. For example, the (equationally axiomatizable) theory of vector spaces over the rationals is an example of a free theory, categorical in every uncountable power, that is ω-free, but not ω-categorical. Tarski formulated the following problem: Is a universal Horn theory that is free in one infinite power necessarily free in all infinite powers? Is it a free theory? Baldwin, Lachlan, and McKenzie, in Baldwin-Lachlan [1973], and Palyutin, in Abakumov-Palyutin-Shishmarev-Taitslin [1973], proved that a countable ω-categorical universal Horn theory is ω1-categorical, and hence categorical in all infinite powers. Thus, it is free in all infinite powers. Givant [1979] showed that a countable ω-free, but not ω-categorical, universal Horn theory is also ω1-categorical, and in fact it is a free theory. Further, he proved that a universal Horn theory, of any cardinality κ, that is κ-free, but not κ-categorical, is necessarily a free theory. Shelah’s research was partially supported by the National Science Foundation and the United States-Israel Binational Science Foundation. This article is item number 404 in Shelah’s bibliography. The authors would like to thank Garvin Melles for reading a preliminary draft of the paper and making several very helpful suggestions. Typeset by AMS-TEX 1 2 STEVEN GIVANT AND SAHARON SHELAH Independently, Baldwin-Lachlan, Givant, and Palyutin all found examples of countable ω1-categorical universal Horn theories that are not ω-free, and of countable ω1and ω-categorical universal Horn theories that are not free theories. Thus, Tarski’s implicit problem remains: find a notion akin to categoricity in power that is regular for universal Horn theories, i.e., if it holds in one infinite power, then it holds in every infinite power. One of the difficulties with the notions of categoricity in power and freeness in power is that they are defined in terms of the cardinality of the universes of models instead of the cardinality of the generating sets. This causes difficulties when trying to work with powers < |T |. Let’s call a model A strictly κ-generated if κ is the minimum of the cardinalities of generating sets of A. We define a theory T to be κ-unique if it has, up to isomorphisms, exactly one strictly κ-generated model. For cardinals κ > |T |, the notions of κ-categoricity, κ-freeness, and κ-uniqueness coincide (in the case when T is universal Horn). When κ = |T |, we have κ-categoricity ⇒ κ-freeness ⇒ κ-uniquenes , but none of the reverse implications hold. Givant [1979], p. 24, asked, for universal Horn theories T , whether κ-uniqueness is the regular notion that Tarski was looking for, i.e., (1) Does κ-uniqueness for one infinite κ imply it for all infinite κ? For countable T , he answered this question affirmatively by showing that ω-uniqueness is equivalent to categoricity in uncountable powers. For uncountable T , he provided a partial affirmative answer by showing that categoricity in power > |T | implies κ-uniqueness for all infinite κ, and is, in turn, implied by κ-uniqueness when κ = |T | and κ is regular. However, the problem whether κ-uniqueness implies categoricity in powers > |T | when ω ≤ κ < |T |, or when ω < κ = |T | and κ is singular, was left open. In this paper we shall prove the following: Theorem. A universal theory T that is κ-unique for some κ > ω is κ-unique for every κ > ω. In particular, it is categorical in powers > |T |. It follows from the previous remarks that a universal Horn theory T which is κ-unique for some κ > ω is κ-unique for every κ ≥ ω. Thus, the only part of (1) that still remains open is the case when T is uncountable and ω-unique. A more general formulation of this open problem is the following: Problem. Is an ω-unique universal theory T necessarily categorical in powers > |T | ? In particular, is a countable ω-unique (or ω-categorical ) universal theory ω1-categorical ? An example due to Palyutin, in Abakumov-Palyutin-Shishmarev-Taitslin [1973], shows that a countable universal theory categorical in uncountable powers need not be ω-unique. In fact, in Palyutin’s example the finitely generated models are all finite, and there are countably many non-isomorphic, strictly ω-generated models. Thus, for universal theories, κ-uniqueness for some κ > ω does not imply ω-uniqueness. To prove our theorem, we shall show that, under the given hypotheses, the theory of the infinite models of T is complete, superstable, and unidimensional, and that all sufficiently large models are a-saturated. Thus, we shall make use of UNIVERSAL THEORIES CATEGORICAL IN POWER AND κ-GENERATED MODELS 3 some of the notions and results of stability theory that are developed in Shelah [1990] (see also Shelah [1978]). We will assume that the reader is acquainted with the elements of model theory and with such basic notions from stability theory as superstability, a-saturatedness, strong type, regular type, and Morley sequence. We begin by reviewing some notation and terminology, and then proving a few elementary lemmas. The letters m and n shall denote finite cardinals, and κ and λ infinite cardinals. The cardinality of a set U is denoted by |U |. The set-theoretic difference of A and B is denoted by A−B. If θ is a function, and ā = 〈a0, . . . , an−1〉 is a sequence of elements in the domain of θ, then θ(ā) denotes the sequence 〈θ(a0), . . . , θ(an−1)〉. We denote the restriction of θ to a subset X of its domain by θ↾X , and a similar notation is employed for the restriction of a relation. A sequence 〈Xξ : ξ < λ〉 of sets is increasing if Xξ ⊆ Xη for ξ < η < λ, and continuous if Xδ = ⋃ ξ<δ Xξ for limit ordinals δ < λ. We use German letters A,B,C, ... to denote models, and the corresponding Roman letters A,B,C, ... to denote their respective universes. If τ(x0, ..., xn−1) is a term in a (fixed) language L for A, and if and ā an n-termed sequence of elements in A, symbolically ā ∈ A, then the value of τ at ā in A is denoted by τ[ā], or simply by τ [ā]. A similar notation is used for formulas. Suppose ā ∈ A and X ⊆ A. The type of ā over X (in A), i.e. the set of formulas in the language of 〈A , x 〉x∈X that are satisfied by ā in the latter model, is denoted by tp(ā, X), or simply by tp(ā, X), when no confusion can arise. The strong type of ā over X (in A), i.e., the set of formulas in the language of 〈A , b 〉b∈A that are almost over X and that are satisfied by ā, is denoted by stp(ā, X), or simply by stp(ā, X). If p(x̄) is a strong type and E ⊆ A is a base for p in A, then p ↾E denotes the set of formulas in p that are almost over E. We write A ⊆ B to express that A is a submodel of B. The submodel of A generated by a set X ⊆ A is denoted by Sg(X), or simply by Sg(X), and its universe by Sg(X). A model is μ-generated if it is generated by a set of cardinality μ, and strictly μ-generated if it is μ-generated, but not ν-generated for any ν < μ. Every model A has a generating set of minimal cardinality, and hence is strictly μ-generated for some (finite or infinite) μ. If X is a generating set of A of minimal cardinality, and Y is any other generating set of A, then there must be a subset Z of Y of power at most |X |+ ω such that Z generates X , and hence also A. A set X ⊆ A is irredundant (in A) if, for every Y $ X we have Sg(Y ) 6= Sg(X). A model A is an n-submodel of B, and B an n-extension of A, in symbols A 4n B, if for every Σn -formula φ(x0, ..., xk−1) in the language of A, and every ā ∈ A, we have A |= φ[ā] iff B |= φ[ā]. A 0-submodel of B is just a submodel in the usual sense of the word, and an elementary submodel—in symbols, A 4 B—is just an n-submodel for each n. We write A ≺ B to express the fact that A is a proper elementary submodel of B, i.e., A 4 B and A 6= B. A theory is model complete if, for any two models A and B, we have A 4 B iff A ⊆ B. For any theory T , we denote by T∞ the theory of the infinite models of T . We shall always use the phrase dense ordering to mean a non-empty dense linear ordering without endpoints. It is well-known that the theory of such orderings admits elimination of quantifiers. Hence, in any such ordering U = 〈U , < 〉, if ā and b̄ are two sequences in U that are atomically equivalent , i.e., that satisfy the same atomic formulas, then they are elementarily equivalent , i.e., they satisfy the 4 STEVEN GIVANT AND SAHARON SHELAH same elementary formulas. Fix a linear ordering V = 〈V , < 〉, and set W = V = ⋃ μ≤ω V . Let ⋖ be the (proper) initial segment relation on W , and, for each μ ≤ ω, let Pμ be the set of elements in W with domain μ, i.e., Pμ = V . There is a natural lexicographic ordering, < , on W induced by the ordering of V : f < g iff either f ⋖ g or else there is a natural number n in the domain of both f and g such that f↾n = g↾n and f(n) < g(n) in V. Take h to be the binary function on W such that, for any f, g in W , h(f, g) is the greatest common initial segment of f and g. We shall call the structure W = 〈W , < , ⋖ , Pμ , h 〉μ≤ω the full tree structure over V with ω+1 levels, or, for short, the full tree over V. Any substructure of the full tree over V that is downward closed , i.e., closed under initial segments, is called a tree over V. A tree is any structure isomorphic to a tree structure over some ordering. A tree U over a dense order V is itself called dense if: (i) for every f in U with finite domain, say n, the set of immediate successors of f in U , i.e., the set of extensions of f in U with domain n+ 1, is densely ordered by < in U, or, put a different way, {g(n) : f ⋖ g} is dense under the ordering inherited from V; (ii) every element in U with a finite domain is an initial segment of an element in U with domain ω. Just as with dense orderings, the theory of the class of dense trees admits elimination of quantifiers. A model A is κ-homogeneous if, for every cardinal μ < κ and every pair ā, b̄ ∈ A of elementarily equivalent sequences, there is an automorphism θ of A taking ā to b̄. It is well-known that, for regular cardinals κ, any model has κ-homogenous elementary extensions (usually of large cardinality). It follows from our remarks above that, if U is a κ-homogeneous dense ordering or tree, and if ā, b̄ ∈ U are atomically equivalent (where μ < κ), then there is an automorphism of U taking ā to b̄. A model A is weakly ω-homogenous provided that, for every n, every pair of sequences ā, ā ∈ A that are elementarily equivalent, and every b ∈ A, there is a b ∈ A such that ā̂ 〈b〉 and ā 〈̂b〉 are elementarily equivalent. It is well known that a countable, weakly ω-homogenous model is ω-homogeneous. Dense orderings and dense trees are always weakly ω-homogeneous. We turn, now, to some notions from stability theory. A model A is a-saturated if, for any strong type p (consistent with the theory of 〈A , a 〉a∈A), if E ⊆ A is a finite base for p, then p↾E is realized in A. We say that A is a-saturated in B, in symbols A 4a B, provided that A 4 B and that, for any strong type p of B based on a finite subset E of A, if p↾E is realized in B, then it is realized in A. A model A is a-prime over a set X if (i) X ⊆ A and A is a-saturated; (ii) whenever B is an a-saturated model elementarily equivalent to A and such that X ⊆ B, then there is an elementary embedding of A into B that leaves the elements of X fixed. (Here, we assume that A and B are elementary substructures of some monster model). As is shown in Shelah [1990], Chapter IV, Theorem 4.18, for complete, superstable theories, the a-prime model over X exists and is unique, up to isomorphic copies over X . We shall denote it by Pra(X). Let A be a model and B,C subsets of A with C ⊆ B. A type p(x̄) over B splits over C if there are b̄, c̄ from B such that tp(b̄, C) = tp(c̄, C), and there is a formula φ(x̄, ȳ) over C such that φ(x̄, b̄) and ¬φ(x̄, c̄) are both in p. We now fix a universal theory T in a language L of arbitrary cardinality. In what follows let L be an expansion of L with built-in Skolem functions, and T ′ any Skolem theory in L that extends T . Without loss of generality, we may assume that, for every term τ(x0, ..., xn−1) of L ′ there is a function symbol f of L of rank UNIVERSAL THEORIES CATEGORICAL IN POWER AND κ-GENERATED MODELS 5 n such that the equation f(x0, ..., xn−1) = τ holds in T . For every infinite ordering U = 〈U , ≤〉 and (complete) Ehrenfeucht-Mostowski set Φ of formulas of L compatible with T , there is a model M = EM (U,Φ) of T —called a (standard) Ehrenfeucht-Mostowski model of T —such that M is generated by U (in particular, U ⊆ M ), and U is a set of Φ-indiscernibles in M with respect to the ordering of U, i.e., if φ(x0, ..., xn−1) ∈ Φ, and if ā ∈ U satisfies ai < aj for i < j < n, then M ′ |= φ[ā]. Shelah [1990], Chapter VII, Theorem 3.6, establishes the existence of certain generalizations of Ehrenfeucht-Mostowski models. Suppose that T is, e.g., nonsuperstable, and that the sequence 〈φn(x̄, ȳ) : n < ω〉 of formulas witnesses this nonsuperstability (see, e.g., Shelah [1990], Chapter II, Theorems 3.9 and 3.14). Then there is a generalized Ehrenfeucht-Mostowski set Φ from which we can construct, for every tree U, a generalized Ehrenfeucht-Mostowski model M = EM (U,Φ) of T . In other words, given a tree U, there a model M of T ′ generated by a sequence 〈āu : u ∈ U〉 of Φ-indiscernibles with respect to the atomic formulas of U; in particular, if w̄ and v̄ in U are atomically equivalent in U, then 〈āw0 , . . . , āwn−1〉 and 〈āv0 , . . . , āvn−1〉 are elementarily equivalent in M . Moreover, for w in P ′ n and v in P ′ ω , we have that M ′ |= φn[āw, āv] iff w ⋖ v. In general, the sequences āu may be of length greater than 1. However, to simplify notation we shall act as if they all have length 1, and in fact, we shall identify āu with u. Thus, we shall assume that M is generated by U and that two sequences from U which are atomically equivalent in U are elementarily equivalent in M. Here are some well-known facts about Ehrenfeucht-Mostowski models. Let U,V be infinite orderings or trees, and M = EM (U,Φ) ,N = EM (V,Φ). (In the case of trees, we must assume that M and N exist.) Fact 1. If θ is any embedding of V into U , then the canonical extension of θ to N ′ is an elementary embedding of N into M. In particular, Fact 2. Any automorphism of U extends to an automorphism of M. Fact 3. If V ⊆ U , then Sg ′ (V ) 4 M , and Sg ′ (V ) is isomorphic to N via a canonical isomorphism that is the identity on V . Fact 4. If U is a linear order, then U is irredundant in M. If U is a tree, then for any element f and subset X of U , if f is not an initial segment of any element of X , then f is not generated by X in M. We shall always denote the reduct of M to L by M, in symbols, M = M↾L. Facts 1, 2, and 4 transfer automatically from M to M. However, M is usually not generated by U . We shall therefore formulate versions of the above facts that apply to Sg(U), and, more generally, to a collection of models that lie between Sg(U) and Sg ′ (U) = M. Definition. Suppose M = EM (U,Φ) , and K ⊆ L − L. (i) For each f ∈ K, let Rf be the relation corresponding to f M , i.e., Rf [a0, . . . , an] iff f[a0, . . . , an−1] = an. Set MK = 〈M , Rf 〉f∈K . (ii) For each V ⊆ U set K∗V = V ∪ {f ′ [ā] : f ∈ K, ā from V } . 6 STEVEN GIVANT AND SAHARON SHELAH Thus, the elements ofK∗V are just the elements of V , together with the elements of M that can be obtained from sequences of V by single applications of f in K. Lemma 1. Let U ,V be dense orderings or dense trees, M = EM (U,Φ) ,N = EM (V,Φ), and K ⊆ L − L. (Thus, in the case of dense trees, we postulate that generalized Ehrenfeucht-Mostowski models for Φ exist.) (i) U is a set of indiscernibles in Sg(K ∗U) for the atomic formulas of Φ (with respect to the atomic formulas of U); in particular, if ā and b̄ are two atomically equivalent sequences in U , then they satisfy the same atomic formulas in Sg(K∗U). (ii) U is a set of indiscernibles in Sg(K∗U) for some complete EhrenfeuchtMostowski set compatible with T , i.e., if ā and b̄ are two atomically equivalent sequences in U , then they are elementarily equivalent in Sg(K∗U). (iii) If θ embeds V into U, then the canonical extension of θ to Sg(K ∗ V ) elementarily embeds Sg(K∗V ) into Sg(K∗U). In particular : (iv) Every automorphism of U extends canonically to an automorphism of Sg(K∗ U). (v) If V ⊆ U, then Sg(K∗V ) 4 Sg(K∗U). The lemma continues to hold if we replace “M ” and “N ” everywhere by “MK ” and “NK ” respectively. Proof. Since satisfaction of atomic formulas is preserved under submodels, part (i) is trivial. Part (i) directly implies (iv), and part (v) implies (iii). Thus, we only have to prove (ii) and (v). We begin with (v). First, assume that U is ω-homogeneous. We easily check that the Tarski-Vaught criterion holds. Here are the details. Suppose ā is an n-termed sequence from Sg(K ∗ V ) satisfying ∃xφ(x, ȳ) in Sg(K ∗U); say, b is an element of Sg(K ∗U) such that 〈b〉̂ ā satisfies φ(x, ȳ). Let c̄ and d̄ be sequences of elements from V and U that generate ā and b respectively. Thus, there are f0, . . . , fk−1 and g0, . . . , gl−1 in K, and a term σ(z̄, w̄) and terms τ0(ū, v̄), . . . , τn−1(ū, v̄) in L such that (adding dummy variables to simplify notation)