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Universal Abelian Groups by Menachem Kojman and Saharon Shelah*

ABSTRACT We examine the existence of universal elements in classes of infinite abelian groups.

The method is defining some invariance of a group relative to a club guessing sequence, a combinatorial tool marketed here to algebraists. We prove, for example: Theorem: For n ≥ 2, there is a purely universal separable p-group in ℵ_{n} if, and only if, 2^{ℵ_{0}} ≤ ℵ_{n}. Theorem: If 2^{ℵ_{0}} ≤ λ and there is μ such that μ^{+} < λ < μ^{ℵ_{0}}, then there is no universal separable p-group of cardinality λ.

§0 Introduction

In this paper “group” will always mean “infinite abelian group”, and “cardinal” and “cardinality” always refer to infinite cardinals and infinite cardinalities.

We call a group G universal for a class of groups K in cardinality λ if every H ∈ K such that |H| ≤ λ is isomorphic to a subgroup of G. The objective of this paper is to examine the existence of universal groups in various well-known classes of infinite abelian groups. We also investigate the existence of purely universal groups, groups with the property that every other group of equal cardinality in the class is isomorphic to one of its pure subgroups.

The main set theoretic tool we use is a club guessing sequence. This is a prediction principle which has enough power to control properties of an infinite object which are defined by looking at all its possible enumerations. However, unlike the diamond, club guessing sequences are proved to exist in ZFC, and therefore using them does not require

* Partially supported by the United States–Israel Binational science foundation. Publication number 455 any additional axioms beyond the usual axioms of ZFC. An important feature of this principle is its usefulness in proving theorems from negations of CH and GCH.

The paper is organized as follows: in Section 1 we define the invariance of a group relative to a club guessing sequence, and show that it is monotone in pure embeddings. In Section 2 we show how to construct various groups with prescribed demands on their invariance. In Section 3 all this machinery is used to investigate the existence of universal groups in classes of torsion groups and classes of torsion free groups. It appears that cardinal arithmetic decides the question of existence of a universal group in quite many cardinals. For example: there is a purely universal separable p-group in ℵ_{n} iff ℵ_{n} ≥ 2^{ℵ_{0}} for all n ≥ 2 (for n = 1 only the “if” part holds).

This paper follows two other papers by the same authors, [KjSh 409] and [KjSh 447], in which the existence of universal linear orders, boolean algebras and models of unstable and stable unsuperstable first order theories were examined using the same method. In a future paper independence results which complement the theorems in these papers and the present one will be presented.

All the abelian group theory one needs here, and more, is found in [Fu], to whose system of notation we adhere. An acquaintance with ordinals and cardinals is necessary, and some familiarity with stationary sets helps. Knowledge of chapter II in [EM] is more than enough. For the reader’s convenience we prepared an appendix in which standard set theoretic terminology and the basics of stationary

sets are reviewed.

Before getting on, we first notice that in every infinite cardinality there are universal groups which are divisible: 0.1 Theorem: In every cardinality there is a universal group, universal p-group for every prime p, universal torsion group and universal torsion-free group.

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Proof: These are, respectively, the direct sum of λ copies of the rational group Q together with λ copies of Z(p^{∞}) for every prime p, the direct sum of λ copies of Z(p^{∞}), the direct sum of λ copies of Z(p^{∞}) for every prime p and the direct sum of λ copis of Q. That these groups are universal, each for its class, follows from the structure theorem for divisible groups and the fact that every group (p-group, torsion group, torsion-free group) is embeddable in a divisible group (p-group, torsion group, torsion-free group) of the same cardinality ([Fu] I, 23 and 24]) _{⌣} · · 0.1

§1 The invariance of a group relative to the ideal id(C)

A fixed assumption in this Section is that λ is a regular uncountable cardinal. We assume the reader is familiar with the basic properties of closed and unbounded sets of λ, and with the definition and basic properties of stationary sets. If not, we recommend a brief look at the appendix, where these are reviewed. 1.1 Definition: For a group G, nG =^{def} {ng : g ∈ G}. Two elements g, h ∈ G are ncongruent if g − h ∈ nG. If g, h are n-congruent, we also say that h is an n-congruent of g. 1.2 Definition:

(1) ([Fuch, p.113]) Let G be group. A subgroup H ⊆ G is a pure subgroup, denoted by H ⊆_{pr} G, if for all natural n, nH = nG ∩ H.

(2) An embedding of groups h : H → G is a pure embedding if its image h(H) is a pure subgroup of G.

1.3 Definition: Suppose that λ is a regular uncountable cardinal and that G is a group of cardinality λ. A sequence G = ⟨G_{α} : α < λ⟩ is called a representation of G (as an increasing and continuous union) iff G_{α} ⊆ G_{α+1} for all α, G_{α} is of cardinality smaller than λ for all α, if α is limit, then G_{α} = ⋃_{β<α} G_{β}, and G = ⋃_{α<λ} G_{α}.

Suppose G = ⟨G_{α} : α < λ⟩ is a given representation of a group G. Suppose c ⊆ λ is

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a set of ordinals, and the increasing enumeration of c is ⟨α_{i} : i < i(∗)⟩. Let g ∈ G be an element. We define a way in which g chooses a subset of c: 1.4 Definition: InvG(g, c) = {α_{i} : α_{i} ∈ C & (∃n)[(∃h ∈ G_{α_{i+1}})((g − h) ∈ nG) & ¬(∃h ∈ G_{α_{i}})((g − h) ∈ nG)]}. We call InvG(g, c) the invariance of the element g relative to the representation G and the set of indices c.

Worded otherwise, the invariance of an element g relative to a given representation and set of indices, is the subset of those indices α_{i} such that by increasing the group G_{α_{i}} to the bigger group G_{α_{i+1}}, for some n an n-congruent for g is introduced.

As the definition of the invariance depends on a representation, one may think that the invariance does not deserve its name. Indeed, given a group G equipped with two respective representations G and G′, it is not necessarily true that for g ∈ G

(1)  InvG(g, c) = InvG′(g, c)

The amendment to this problem is working with a club guessing sequence C = ⟨c_{δ} : δ ∈ S⟩. The idea is as follows: for any pair of representations G and G′ of the same group G there is a club of indices E ⊆ λ such that for every α ∈ E, G_{α} = G′_{α}. So if we chose our set c in the definition of invariance to be consisted only of such “good” α-s, namely if c ⊆ E, then it would not matter according to which representation we work. But we cannot choose a set c which will be a subset of every club E resulting from some pair of representations. What we can do is find a sequence of c-s with the property that for every club E ⊆ λ, stationarily many of them are susets of E. Thus we will be able to define an invariance that will prove to be independent of a particular choice of a representation. Here is the precise formulation of this: 1.5 Definition: A sequence ⟨c_{δ} : δ ∈ S⟩, where S ⊆ λ is a stationary set, c_{δ} ⊆ δ and δ = sup c_{δ} for every δ, is called a club guessing sequence if for every club E ⊆ λ the set {δ ∈ S : c_{δ} ⊆ E} is a stationary subset of λ.

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The theorems asserting the existence of club guessing sequences will be quoted later. A club guessing sequence C = ⟨c_{δ} : δ ∈ S⟩ gives rise to an ideal id(C) over λ, the guessing ideal. 1.6 Definition: Suppose that C = ⟨c_{δ} : δ ∈ S⟩ is a club gussing sequence. We define a proper ideal (namely a notion of “small” sets) id(C) as follows:

A ∈ id(C) ⇔ A ⊆ λ & ∃E ⊆ λ, E club, & ∀δ ∈ E ∩ S, c_{δ} ⊈ E.

So a set of δ-s is small if there is a club E which it fails to guess stationarily often, namely there is no δ ∈ A such that c_{δ} ⊆ E. 1.7 Lemma: If C is a club guessing sequence as above, then id(C) is a proper, λ complete ideal. Proof: That id(C) is proper means that it does not contains every subset of λ. Indeed, S ∉ id(C), as it guesses every club. That id(C) is downward closed is immediate from the definition. Suppose, finally, that A_{i}, i < (∗), i(∗) < λ are less than λ sets in the ideal. We wish to show that their union A =^{def} ⋃_{i<i(∗)} A_{i} is in the ideal. Pick a club E_{i} for every i < i(∗) which demonstrates that A_{i} ∈ id(C), namely δ ∈ A_{i} ⇒ c_{δ} ⊈ E_{i}. Let E ⋂_{i<i(∗)} E_{i}. E is a club. We show that E demonstrates that A ∈ id(C): Suppose that δ ∈ A. Then there is some i < i(∗) such that δ ∈ A_{i}. Therefore c_{δ} ⊈ E_{i}. But E ⊆ E_{i}, so necessarily c_{δ} ⊈ E. _{⌣} · · 1.7

We adopt the term “for almost every δ in S”, by which we mean “for all δ ∈ S except for a set in id(C)”. 1.8 Lemma: Suppose that C = ⟨c_{δ} : δ ∈ S⟩ is a club guessing sequence on S ⊆ λ. Suppose that G and G′ are two representations of a group G of cardinality λ. Then for almost every δ ∈ S, (1) holds for every g ∈ G. Proof: : The set of α < λ for which G_{α} = G′_{a} is a club. Let us denote it by E. If for some δ, c_{δ} ⊆ E holds, then for every g ∈ G it is true that InvG(g, c_{δ}) = InvG′(g, c_{δ}). But as C

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is a club guessing sequence, by definition, for almost every δ, c_{δ} ⊆ E. _{⌣} · · 1.8

We define now the invariance of a group. 1.9 Definition: Suppose that C is a club guessing sequence and that G is a representation of a group G of cardinality λ. Let

(1) P_{δ}(G, C) = {InvG(g, c_{δ}) : g ∈ G}

(2) INV(G, C) = [⟨P_{δ}(G, C) : δ ∈ S⟩]_{id(C)}

The P_{δ} is the collection of all subsets of c_{δ} which serve as the invariance of some g ∈ G. The second item should read “the equivalence class of the sequence of P_{δ} modulo the ideal id(C)”, where two sequences are equivalent modulo an ideal if the set of coordinate in which the sequences differ is in the ideal. 1.10 Lemma: The definition of INV(G, G) does not depend on the representation G. Proof: Suppose that G′ is another representation. There is a club E such that for every α ∈ E, G_{α} = G′_{α}. Therefore for every δ such that c_{δ} ⊆ E and every g ∈ G, InvG(g, c_{δ}) = InvG′(g, c_{δ}). This means that for every δ such that c_{δ} ⊆ E, P_{δ}(G, C) = P_{δ}(G′, C). But for almost all δ it is true that c_{δ} ⊆ E, therefore the sequqnces ⟨P_{δ}(G, C) : δ ∈ S⟩ and ⟨P_{δ}(G′, C) : δ ∈ S⟩ are equivalent modulo id(C). _{⌣} · · 1.10

We have exhibited here a general method of defining well an invariance of a group with the help of representing it as an increasing union of smaller groups. This depends, though, on the existence of a club guessing sequence! Strangely enough, we can prove the existence of club guessing sequences for regular uncountable cardinals λ for all such cardinals except ℵ_{1}. We believe that this method of making definitions may have further algebraic applications beyond those made in this paper.

Let us now quote the relevant theorems which assert the existence of club guessing sequences:

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1.11 Theorem: If μ and λ are cardinals, μ^{+} < λ and λ is regular, then there is a club guessing sequence ⟨c_{δ} : δ ∈ S⟩ such that the order type of each c_{δ} is μ.

By [Sh-g] Ch. III: [Sh 365] or see the appendix to [KjSh 409] for proofs of this. Preservation of INV under pure embeddings

We procced to show that INV is preserved, in a way, under pure embeddings. 1.12 Lemma: Suppose that H and G are groups of cardinality λ and that H and G are representations. Suppose that C is a club guessing sequence on S ⊆ λ. If h : H → G is a pure embedding, then for almost every δ ∈ S P_{δ}(H, C) ⊆ P_{δ}(G, C). Proof:

Suppose for simplicity that H ⊆_{pr} G, namely that the embedding is the identity function. The set E_{1} =^{def} {α : H ∩ G_{α} = H_{α}} is a club. Define for every natural number n a function f_{n}(y) on G as follows:

f_{n}(y) = \{ \begin{matrix}some x ∈ \{x : x ∈ H & (x + y) ∈ nG\} & if \{x : x ∈ H & (x + y) ∈ nG\} ≠ ∅ \\ 0 & otherwise \end{matrix}

There is a club E_{2} such that G_{α} is closed under f_{n} for all n for every α ∈ E_{2}. E = E_{1} ∩ E_{2} is a club. 1.13 Claim: Suppose that h ∈ H and that α ∈ E. Then h has an n-congruent in G_{α} (in the sense of G) iff h has an n-congruent in H_{α} (in the sense of H). Proof: One direction is trivial. Suppose, then, that there is an n-congruent g ∈ G_{α}. Let h′ = f_{n}(g). By the definition of f_{n}, h′ + g ∈ nG_{α}; also h − g ∈ nG. Therefore h − h′ ∈ nG. As H ⊆_{pr} G, h − h′ ∈ nH, and therefore h′ is an n-congruent of h in the sense of H. _{⌣} · · 1.13

The proof of the Lemma follows now readily: For almost every δ ∈ S it is true c_{δ} ⊆ E. Therefore for every such δ, every h ∈ H and every n, h has an n-congruent in H_{α} iff h has an n-congruent in G_{α}. Therefore InvH(h, c_{δ}) ⊆ InvG(h, c_{δ}). _{⌣} · · 1.12

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§2 Constructing groups with prescribed INV

We need to be able to construct a group G such that for a prescribed list of sets ⟨A_{δ} : δ ∈ S⟩, A_{δ} ∈ P_{δ}(G, C). In this Section we construct several groups with such prescribed demands on their INV. These will be used in the next Section for showing that in certain cardinals universal groups do not exist. The technique in all the constructions will be attaching to a simply defined group points from a topological completion of the group.

a. Constructions of p-groups 2.1 Theorem: If λ is a regular uncountable cardinal, C = ⟨c_{δ} : δ ∈ S⟩ is a club guessing sequence and A_{δ} ⊆ c_{δ} is a given set of order type ω, then there is a separable p-group G of cardinality λ and representation G such that A_{δ} ∈ P_{δ}(G, C) for every δ ∈ S. 2.2 Remark: This implies by Theorem 1.12 that for every separable p-group G′ of cardinality λ into which G can be purely embedded and a representation G′, for almost

every δ, A_{δ} ∈ P_{δ}(G′, C)

Proof: For every n, let B_{n} = ⨁_{η∈^{n}λ} A_{η} where A_{η} is a copy of Z_{p^{n}} with generator a_{η}. Let G^{0} = ⨁_{n} B_{n}, and let G^{1} be the torsion completion of G^{0}. G^{1} may be identified with all sequences (x_{1}, x_{2}, ⋯) where x_{n} ∈ B_{n} and such that there is a (finite) bound to {o(x_{n})}_{n}. For details see [Fuchs II,14–21]. The group we seek lies between G^{0} and G^{1}, and is a pure subgroup of G^{1}.

Let us make a simple observation:

(1) If x = (x_{1}, x_{2}, ⋯) and y = (y_{1}, y_{2}, ⋯) belong to G^{1} and x − y ∈ p^{n}G^{1}, then x_{i} = y_{i} for all i ≤ n.

Proof: Let z_{i} = x_{i} − y_{i}. z_{i} ∈ B_{i}. As B_{i} ∩ p^{n}G^{1} = 0 for i ≤ n, we are done.

For every δ ∈ S let ⟨α_{n}^{δ} : n ∈ ω⟩ be the increasing enumeration of A_{δ}. Denote by η_{n}^{δ} the sequence ⟨α_{1}^{δ}, ⋯, α_{n}^{δ}⟩. Let b_{δ}^{0} ∈ G^{1} be (x_{1}^{δ}, x_{2}^{δ}, ⋯) where x_{n}^{δ} = p^{n−1}a_{η_{n}^{δ}}. So x_{n} is of order p and height n − 1. Consequently, b_{δ}^{0} is of order p. Let us denote

x_{k}^{δ} ; p^{n} =^{def} \{ \begin{matrix}p^{k−n−1}a_{η_{k}^{δ}} & if n < k \\ 0 & otherwise \end{matrix}

and also let 0 ; p^{n} =^{def} 0. Let b_{δ}^{n} = (⋯, x_{k}^{δ} ; p^{n} ⋯). Let G be the subgroup of G^{1} generated by G^{0} together with {b_{δ}^{n} : δ ∈ S, n < ω}. Having defined G, let us specify a representation G. For every i < λ let G_{i} be ⟨{a_{η} : η ∈ ^{<ω}i} ∪ {b_{δ}^{n} : δ ∈ S, δ < i, n < ω}⟩. 2.3 Claim: InvG(b_{δ}^{0}, c_{δ}) = A_{δ}. Proof: We should show that the set of indices i with the property that in G_{i+1} some congruent of b_{δ}^{0} appears coincides with A_{δ}. Suppose first, then, that i = α_{n} for some n. (x_{1}^{δ}, ⋯, x_{n}^{δ}, 0, 0, ⋯) is clearly a p^{n}-congruent of b_{δ}^{δ}, as b_{δ}^{0}−(x_{1}^{δ}, ⋯, x_{n}^{δ}) = p^{n}b_{δ}^{n}. Conversely, suppose that i < α_{n} and suppose to the contrary that there is some y = (y_{1}, y_{2}, ⋯) ∈ G_{i} such that b_{δ}^{0} − y ∈ P^{n}G. Then y_{i} = x_{i} for all i ≤ n by (1). But y ∈ G_{i} implies that y_{n} ∈ G_{i} — a contradiction to i < α_{n}. _{⌣} · · 2.3,2.1

We shall prove now a strengthening of the previous theorem: We shall construct a separable p-group such that the demands on its INV will hold in any separable p-groups of the same cardinality extending it — not necessarily as a pure extension. The Price we pay for this is that the cardinality of the group we construct must be at least 2^{ℵ_{0}}. 2.4 Theorem: Suppose that λ ≥ 2^{ℵ_{0}} is a regular cardinal, and that C = ⟨c_{δ} : δ ∈ S⟩ is a club guessing sequence . Suppose also that for every δ a subset A_{δ} ⊆ c_{δ} of order type ω is given. Then there exists a separable p-group G of cardinality λ and a representation G of G with the property that for every separable p-group H of cardinality λ (with representation H) and embedding φ : G → H, for almost every δ, A_{δ} ∈ P_{δ}(H, C). Proof: We define G^{0} = ⨁_{n} B_{n}, where B_{n} = ⨁_{η∈^{<ω}λ} A_{η}^{n} is a direct sum of isomorphic copies of Z_{p^{n}}. We fix a_{η}^{n} as a generator of A_{η}^{n}. G^{1} is the torsion completion of G^{0}. The group G we seek lies between G^{0} and G^{1}.

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Enumerate in the sequence ⟨Γ_{i} : i < 2^{ℵ_{0}}⟩ all infinite subsets of ω. For every i < 2^{ℵ_{0}} we throw into G an element b_{δ,0}^{i}, (which will have A_{δ} as its invariance). Let ⟨η(n) : n < ω⟩ be the increasing enumeration of A_{δ}. We define b_{δ,0}^{i} = ⟨x_{1}^{δ}, x_{2}^{δ}, ⋯⟩, an element in G^{1}, as follows:

x_{n}^{δ} = \{ \begin{matrix}0 & if n ∉ Γ_{i} \\ p^{n−1}a_{η k}^{n} & if n is the k-th member of Γ_{i} \end{matrix}

Let

x_{l}^{δ} ; p^{n} =^{def} \{ \begin{matrix}p^{l−n−1}a_{η k}^{n} & if x_{l}^{δ} = p^{l−1}a_{η k}^{l} and l > n \\ 0 & otherwise \end{matrix}

Let G be the subgroup of G^{1} generated by G^{0} ⋃{b_{δ,n}^{i} : δ ∈ S, i < 2^{ℵ_{0}}, n < ω}, where

b_{δ,n}^{i}, for n > 0, is (⋯, x_{k}^{δ} ; p^{n} , ⋯). Let G_{α} = ⟨{a_{ν}^{n} : n < ω, ν ∈ ^{n}α} ∪ {b_{δ,n}^{i} : δ ∈ S, δ <

α, n < ω}⟩.

Suppose now that φ : G → H is an embedding, H any separable p-group of cardinality λ. By picking a basic subgroup H_{0} = ⨁_{n} C_{n} of H, we may identify elements of H with the bounded sequences ⟨y_{1}, y_{2}, ⋯⟩ such that y_{n} ∈ C_{n}. Fix any representation H of H. Clearly, the set E = {α < λ : φ[G_{α}] ⊆ H_{α} & φ^{−1}[H_{α}] = G_{α}} is a club of λ. Therefore it is enough to show that for every δ ∈ S such that c_{δ} ⊆ E, A_{δ} ∈ P_{δ}(H, C).

Let us look, then, at one such δ. Observe that if a ∈ G is of height k, then the height of φ(a) is at least k. Define by induction x_{n} as follows: x_{1} = a_{η 1}^{1}, and x_{n+1} = p^{h(φ(x_{n}))}a_{η (n+1)}^{h(φ(x_{n}))+1}.

There exists some index i < 2^{ℵ_{0}} such that b_{δ,0}^{i} = ⟨x_{1}^{δ}, 0, 0, ⋯, x_{2}, 0 ⋯⟩ is such that x_{k}^{δ} is the k-th non-zero element of b_{δ}^{i,0}. Let us see that InvH(φ(b_{δ,0}^{i}), c_{δ}) = A_{δ}. Let α_{k} be the k-th member of A_{δ}. Let n = h(φ(x_{k})) + 1. We shall see that in H_{α_{k+1}} a p^{n}- congruent to φ(b_{δ,0}^{i}) is introduced. First, φ(x_{1}, 0, ⋯ 0, x_{k}) is clearly a p^{n}-congruent, for φ(b_{δ,0}^{i} − (x_{1}, 0 ⋯ 0, x_{k})) = p^{n}φ(b_{δ,n}^{i}). As α_{k}, α_{k+1} ∈ E, φ((x_{1}, 0 ⋯, x_{k})) ∈ G_{α_{k+1}} \ G_{α_{k}}. On the other hand, h(φ((x_{1}, 0, ⋯, 0, x_{k}))) < n, so as a p^{n}-congruent is unique modulo

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p^{n}H, there is no p^{n}-congruent of φ(b_{δ,0}^{i}) in H_{α_{k}}. _{⌣} · · 2.4 b. Constructions of torsion-free groups

We start by constructing a torsion-free homogeneous group of a given type t = (∞, ⋯, ∞, 0, ∞ ⋯). We recall that a characteristic χ(g) of an element g ∈ G is the sequence (k_{1}, k_{2}, ⋯) where k_{l} is the p-height of g for the l-th prime. The height can be ∞. A type t is an equivalence class of characteristics modulo the equivalence relation of having only a finite difference in a finite number of coordinates. A homogeneous group is a group in which all elements have the same type. We call a type t a p-type if t = (∞, ⋯, ∞, 0, ∞ ⋯) where the only coordinate in which there is 0 is the number of p in the list of primes. 2.5 Theorem: For every uncountable and regular cardinal λ, a club guessing sequence C = ⟨c_{δ} : δ ∈ S⟩ and given sets A_{δ} ∈ c_{δ}, each A_{δ} of order type ω, there is a homogeneous group G of cardinality λ with p-type t and a representation G such that for every δ ∈ S,

A_{δ} ∈ P_{δ}(G, C).

2.6 Remark: This means that for every pure extension G′ of G, for almost every δ ∈ S,

A_{δ} ∈ P_{δ}(G′, C).

Proof: This proof resembles the proof of Theorem 2.1. Let G^{0} = ⨁_{λ} Q_{p} (where Q_{p} is the group or rationals with denominators prime to p). We index the isomorphic copies of Q_{p} by η ∈ ^{n}λ and fix a_{η}, an element a_{η} of characteristic (∞, ⋯, ∞, 0, ∞ ⋯) in the η-th copy of Q_{p}. Let G^{1} be the completion of G^{0} in the p-adic topology. Let ⟨η^{δ}(n) : n < ω⟩ be the increasing enumeration of A_{δ}, and let η_{n}^{δ} = ⟨η^{δ}(0), ⋯, η^{δ}(n − 1)⟩. Let b_{δ,n} = ∑_{k} p^{k−n}a_{η_{k}}. The rest is as in the proof of Theorem 2.1. _{⌣} · · 2.5

Working above the continuum, we choose to construct two types of groups: homogeneous groups and slender groups. 2.7 Theorem: Suppose that λ > 2^{ℵ_{0}} is a regular cardinal, and that C = ⟨c_{δ} : δ ∈ S⟩

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is a club guessing sequence. Suppose also that for every δ ∈ S a subset A_{δ} ⊆ c_{δ} of order type ω is given. Then there exists a homogeneous group G of p-type t of cardinality λ and a representation G of G with the property that for every reduced torsion-free group H of cardinality λ (with representation H) and embedding φ : G → H, for almost every

δ, A_{δ} ∈ P_{δ}(H, C).

Proof: With the same adaptations as in the previous proof, this resembles the proof of Theorem 2.4. _{⌣} · · 2.7

We recall that a a slender group is a group into which there is no homomorphism of P = ∏_{ℵ_{0}} that sends an infinite number of e_{n} to non-zero images. Slender groups are reduced, and a group of cardinality smaller than 2^{ℵ_{0}} is slender iff it is reduced (see [Fu], II, 94). The demands on INV will be preserved under any embedding into a reduced group. 2.8 Theorem: Suppose that λ > 2^{ℵ_{0}} is a regular cardinal, and that C = ⟨c_{δ} : δ ∈ S⟩ is a club guessing sequence. Suppose also that for every δ a subset A_{δ} ⊆ c_{δ} of order type ω is given. Then there exists a slender group G of cardinality λ and a representation G of G with the property that for every reduced torsion-free group H of cardinality λ (with representation H) and embedding φ : G → H, for almost every δ, A_{δ} ∈ P_{δ}(H, C). Proof: For every pair ⟨η, ρ⟩ such that η ∈ ^{<ω}λ, ρ ∈ ^{<ω}ω and lg η = lg ρ, let x_{⟨η,ρ⟩} be a free generator. Let G^{0} =^{def} ⨁⟨x_{⟨η,ρ⟩}⟩. Let G^{1} =^{def} Ĝ^{0}, the Z-adic completion of G^{0}. The group we seek lies between G^{0} and G^{1}. Let Δ be the set of increasing ω-sequences of integers ρ = ⟨ρ(0), ρ(1), ⋯⟩ such that n!ρ(n)|ρ(n + 1). For every δ ∈ S let η_{δ} enumerate the members of A_{δ}.

For every δ ∈ S we throw into G the elements b_{η_{δ},ρ,k} for all ρ ∈ Δ and natural k,

where b_{η_{δ},ρ,0} = ∑_{l<ω} ρ(l)x_{η_{δ} l,ρ l} and b_{η_{δ},ρ,k+1} = ∑_{k<l<ω} ρ(l) ; ρ(k)x_{η l,ρ l}.

Let G_{α} = ⟨{x_{⟨η,ρ⟩} : η ∈ ^{<ω}α, ρ ∈ ^{<ω}ω, lg η = lg ρ} ∪ {b_{η_{δ},ρ,k} : δ ∈ S, δ < α, ρ ∈ Δ, k < ω, }⟩. Let G = ⋃_{α<λ} G_{α}.

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For an element g ∈ G we define the support supp g in the following manner: The quotient G/n!G is naturally isomorphic to ⨁ _{n!}x_{⟨η,ρ⟩}. Therefore we can define supp_{n}g as the support in this direct sum of [g]_{n!G}, (where by “support” of a sum we mean the finite set of x_{⟨η,ρ,⟩} with non zero coefficients in this sum). We let supp g =^{def} ⋃_{n} supp_{n}G. Trivially, an element g ∈ G is in the closure of ⨁_{x_{⟨η,ρ,⟩}∈supp g} x_{⟨η,ρ,⟩}. Also, supp (g+h) ⊆ supp g∪supp h. We notice also the following obvious fact: 2.9 Fact: If g ∈ G then there is a finite set {b_{η_{δ^{1}} ,ρ^{1},k^{1}}^{1}, ⋯, b_{η_{δ^{l}} ,ρ^{l},k^{l}}^{l}} such that supp g is almost included in ⋃_{0<i<l} supp b_{η_{δ^{i}} ,ρ^{i},k^{i}}^{i}, where “almost” means “except for finitely many elements”.

We should show that G is slender. We prove the following 2.10 Claim: For every set of non-zero elements {y_{n} : n < ω} such that n!|y_{n}, there is a sequence of integer coefficients a_{n} such that the series ∑ a_{n}y_{n} does not converge in G.

First, let us point out that this claim implies the slenderness of G. Suppose that φ : P → G is a homomorphism satisfying that (without loss of generality) for all n, φ(e_{n}) ≠ 0. Then let y_{n} = n!φ(e_{n}). Apply the claim to obtain a member x ∈ P for which φ(x) is not defined. Proof: (of Claim) For a finite set B ⊆ ^{ω}λ ×^{ω}ω, we let A_{B} = {x_{⟨η n,ρ n⟩} : ⟨η, ρ⟩ ∈ B, n < ω}. For A = A_{B} for some finite B as above, we define an endomorphism T_{A} : G → G as follows:

T_{A}(x_{⟨η,ρ⟩}) = \{ \begin{matrix}x_{⟨η,ρ⟩} & if ⟨η, ρ⟩ ∈ A \\ 0 & otherwise \end{matrix}

The image of T_{A} is a closed and countable subgroup of G.

Let y_{n} ∈ n!G be given for all n < ω. y_{n} = a_{0}^{n} + ∑_{l<l(n)} a_{l}^{n}b_{η_{δ} _{l}^{n},ρ_{l}^{n},k_{l}^{n}}, where a_{0}^{n} ∈ G_{0}.

We distinguish two cases:

First case: There is a finite set B as above such that for infinitely many n, supp y_{n}∩A_{B}

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is not empty. Let A denote A_{B}.

We assume for simplicity that for every n, T_{A}(y_{n}) ≠ 0. If every infinite sum ∑ a_{n}y_{n} existed in G, then the image under T_{A} of every sum would exist in ImT_{A}. But this is a contradiction to The countability of ImT_{A}.

Second case: For every finite set B ⊆ ^{ω}λ × ^{ω}ω, there are only finitely many n for which supp y_{n} ∩ A_{B} is not empty. We may assume by passing to a subsequence that the {supp y_{n} : n < ω} is a collection of pairwise disjoint sets.

We choose coefficients c_{n} by induction: let c_{n+1} be so big, that for all ∑_{m≤n} c_{m}y_{m} ∉ c_{n+1}G. Call the resulting sum g. The choice of the coefficients guarantees that that supp g is not almost included in any finite union of supports of b_{η_{δ},ρ,k}, and therefore by Fact 2.9 is not in G. _{⌣} · · 2.10

Suppose now that H is a representation of a torsion-free group of cardinality λ and that φ : G → H is a monomorphism. It is enough to show that for every δ wich satisfies that α ∈ c_{δ} ⇒ φ[G_{α}] ⊆ H_{α} ∧ φ^{−1}[H_{α}] = G_{α}, A_{δ} ∈ P_{δ}(H, C). We define by induction an ω-sequence ρ ∈ Δ. Let ρ(0) = 1. Suppose that (ρ(0), ⋯, ρ(n − 1)) are defined, and for convenience denote by ρ n this finite sequence. Let ρ(n) be so big that ∑_{m<n} ρ(m)φ(x_{η_{d} m,ρ m} ∉ ρ(n)G and n!ρ(n − 1)|ρ(n). The sequence ρ just defined is clearly in Δ. We let the reader verify

that InvH(φ(b_{η_{δ},ρ,o}), c_{d}) = A_{δ}. _{⌣} · · 2.8

§3 The Theorems a. The Universality Spectrum of Torsion groups

There is universal torsion group in λ iff there is a universal p-group in λ for every prime p. We therefore may focus on p-groups alone. There is a universal divisible p-group in λ, the group ⨁_{λ} Z(p^{∞}), therefore the first interesting question to ask in torsion groups is whether there is a universal reduced p-group. Here the answer is “no”: 3.1 Theorem: If λ is an infinite cardinal (not necessarily regular, not necessarily un-

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countable) then ther is no universal reduced p-group in λ. Proof: There are p-groups of cardinality λ of Ulm length σ for every ordinal σ < λ^{+}. As u(A) ≤ u(B) whenever A ⊆ B, and u(A) < λ^{+}, for every group of cardinality λ, no p-group of cardinality λ can be universal. _{⌣} · · 3.1

We put a further restriction on the class of p-groups, by demanding that the Ulm length of a group be at most ω.^{1} We restrict ourselves then to the class of separable p-groups. On this class see [Fu] vol II, chapter XI. b. Universal separable p-groups

In this Section we investigate the universality spectrum of the class or separable pgroups. 3.2 Theorem: If λ = λ^{ℵ_{0}} then there is a purely universal separable p-group in λ. Proof: Let B = ⨁ B_{n} where B_{n} = ⨁_{λ} Z_{p^{n}}. The torsion completion of B, denoted by G, is of cardinality |B|^{ℵ_{0}} = λ^{ℵ_{0}} = λ and is puely universal in λ. To see this let A be any separable p-group of cardinality λ, and let B_{A} be its basic subgroup. B_{A} is purely embeddable in B, and this gives rise to a pure embedding of A into G. _{⌣} · · 3.2

We see then, that for every n such that ℵ_{n} ≥ 2^{ℵ_{0}} there is a purely universal separable p-group in ℵ_{n}. Therefore CH implies that in every ℵ_{n} there is a purely universal separable p-group. It is not uncommon that CH adds enough information to decide questions in algebra. It is much less common, though, that a negation of CH adds information. (This, we believe, is the reason that although CH and ¬CH are equally consistent with ZFC, it is CH that has become popular among mathematicians). We shall see now that also a negation of CH may be useful.

1 One can make finer distictions here by considering the class of all p-groups of Ulm length which is bounded by an ordinal σ. But we do not do this here.

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3.3 Theorem: If λ is regular and uncountable, and ℵ_{1} < λ < 2^{ℵ_{0}}, then there is no purely universal separable p-group. Proof: Let G be any separable p-group of cardinality λ, and we shall show that it is not purely universal. We fix some presentation G, and by Theorem 1.11 choose some club guessing sequence C = ⟨c_{δ} : δ ∈ S⟩ where S is a stationary subset of λ. We may assume that the order type of each c_{δ} is ω. As |G| = λ, for every δ ∈ S, |P_{δ}(G, C)| =

|{InvG(g, c_{δ}) : g ∈ G}| ≤ λ. As |P(c_{δ})| = 2^{ℵ_{0}} > λ, there is some A_{δ} ⊆ c_{δ}, A_{δ} ∉ P_{δ}(g, c).

Having thus chosen a_{δ} for every δ ∈ s, we invoke Theorem 2.1 to obtain a group H and a representation H such that for every δ ∈ S, A_{δ} ∈ P_{δ}(H, C). If H were purely embedded in G, then by theoren 1.12, for almost every δ in S, A_{δ} would belong to P_{δ}(G, C). This clearly contradicts the choice of A_{δ}. Therefore H is not purely embeddable in G, and G is therefore not purely universal. _{⌣} · · 3.3

3.4 Corollary: For n ≥ 2, there is a purely universal separable p-group in ℵ_{n} if, and only if, 2^{ℵ_{0}} ≤ ℵ_{n}.

Next we handle cardinals which are bigger than 2^{ℵ_{0}}. 3.5 Theorem: λ > 2^{ℵ_{0}} is regular and there is some μ such that μ^{+} < λ < μ^{ℵ_{0}} then there is no universal separable p-group in λ. Proof: By μ^{+} < λ and Theorem 1.11, we may pick some club guessing sequence C = ⟨c_{δ} : δ ∈ S⟩ where S is a stationary set of λ and otp c_{δ} = μ. Suppose G is a given separable p-group. We will show that G is not universal by presenting a separable p-group H of cardinality λ which is not embeddable in G. We choose a representation G of G and observe that |P_{δ}(G, C)| ≤ λ for every δ. As μ^{ℵ_{0}} > λ, there is some A_{δ} ⊆ c_{δ}, of order type ω, which dose not belong to P_{δ}(G, C). By Theorem 2.4, there is a group H of cardinality λ such that for every embedding φ : H → G, for almost every δ, A_{δ} ∈ P_{δ}(G, C). This can hold only emptily, that is, if there are no such embeddings, because A_{δ} was chosen such

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that A_{δ} ∉ P_{δ}(G, C) _{⌣} · · 3.5

Examining the conditions on λ in the last three theorems, one discovers that for purely universal separable p-groups there are cases for regular λ which are not covered by them: If cfλ = λ < λ^{ℵ_{0}}, there is some first μ < λ for which μ^{ℵ_{0}} > λ. This μ must be of cofinality ℵ_{0}, or else μ^{ℵ_{0}} = ⋃_{α<μ} α^{ℵ_{0}} ≤ λ, as for every α < μ, α^{ℵ_{0}} ≤ λ. So there are two cases: μ^{+} < λ and μ^{+} = λ. The first case is covered by 3.2 (if μ = ℵ_{0}) or by 3.5, (if μ > 2^{ℵ_{0}}). The second case is not covered here. The authors plan to write a paper with consistency results, in which it will be shown that in cardinals which fall in the second case the existence of a universal separable p-group is consistent (namely, one cannot prove that there is no such group).

Also, Theorem 3.5 eliminates the existence of universal separable p-groups in λ, μ^{+} < λ < μ^{ℵ_{0}}, while Theorem 3.2 eliminates only the existence of purely-universal separable pgroups in λ, ℵ_{1} < λ < 2^{ℵ_{0}}. A consistency result showing that in a regular cardinal λ < 2^{ℵ_{0}} there may exist a universal separable p-group will also be presented. We shall add here a theorem that shows that the existence of a universal separable p-group in cardinality λ, when λ < 2^{ℵ_{0}}, is equivalent to the existence of a universal separable p-group with countable dimensions. 3.6 Definition: For a cardinal λ let G_{λ} = ⨁_{n} B_{n}^{λ} where B_{n}^{λ} = ⨁_{i<λ} (p^{n})x_{i}^{n}. Let Ĝ_{λ} be the torsion completion of G_{λ}. The form of an element in Ĝ_{λ} is ∑_{n<ω,i<λ} c_{n,i}x_{i}^{n} where for each n only a finite number of c_{n,i} are not zero, and such that there is a common bound p^{k} to the orders of all the terms in the sum.

It is clear that every separable p-group of cardinality λ is realized as a pure subgroup of Ĝ_{λ}.

When λ ≤ 2^{ℵ_{0}} we observe the following fact: 3.7 Fact: If λ ≤ 2^{ℵ_{0}} than Ĝ_{λ} is embeddable into Ĝ_{ℵ_{0}}

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Proof: For brevity let us denote G_{ℵ_{0}} as G_{0}. G_{0} = ⨁ B_{n}^{0}, where B_{n}^{0} is a direct sum of ℵ_{0} many copies of (p^{n}). As it makes no difference which countable index set we use to enumerate the summands of B_{n}^{0}, let B_{n}^{0} = ⨁_{η∈^{n}2,m<ω} (p^{n})y_{η}^{n,m}. As λ < 2^{ℵ_{0}} we can find λ many sequences η_{i}^{n} ∈ ^{ω}2 (n < ω, i < λ). We define a homomorphism φ : Ĝ_{λ} → G_{0} by specifying the images of each x_{i}^{n}. So φ(x_{i}^{n}) = ∑_{n≤m} p^{m−n}y_{η_{i}^{m} n}^{m,n}. We extend φ to all of Ĝ_{λ} by continuity, namely φ(∑_{n,i} c_{i,n}x_{i}^{n}) = ∑_{n,i} c_{i,n}φ(x_{i}^{n}).

We have yet to show that φ is 1-1. Suppose, then, that ∑_{n,i} c_{i,n}x_{i}^{n} is a non zero element of Ĝ_{λ}. We shall see that its image under φ is not zero. Suppose that n_{0}, i_{0} are such that c_{n_{0},i_{0}}x_{i_{0}}^{n_{0}} ≠ 0. Let all i for which c_{i}^{n_{0}} ≠ 0 be in the set {i_{0}, ⋯, i_{k}}. There is some l(∗) > n_{0} such that for every 0 < j ≤ k, η_{i_{0}} l(∗) ≠ η_{i_{j}} l(∗). Now φ(∑_{n,i} c_{i,n}x_{i}^{n}) =

∑_{n,i} c_{i,n} ∑_{n≤m} p^{m−n}y_{η_{i} n}^{m,n} = ∑_{i,n≤m} c_{i,n}p^{m−n}y_{η_{i} n}^{m,n}. As c_{n_{0},i_{0}}x_{i_{0}}^{n_{0}} ≠ 0, p^{n_{0}} |̸c_{i_{0}}^{n_{0}}, and it follows

that c_{n_{0},i_{0}}p^{l(∗)−n_{0}}y_{η_{i_{0}}}^{l(∗),n_{0}} l(∗) ≠ 0. But this term appears only once in the infinite sum above, and cannot be cacelled, as all other therms are in a direct summand of G_{0} not containing it. We conclude that the image is not zero. _{⌣} · · 3.7

In light of the last Theorem, if λ < 2^{ℵ_{0}}, then the existence of a separable p-group of cardinality λ in which every subgroup of Ĝ_{0} of cardinality λ is embedded, implies the existence of a universal separable p-group in λ (every separable p-group of cardinality λ is embedded in Ĝ_{λ}; Ĝ_{λ} is embedded in Ĝ_{0}; so the composition of both embeddings embedds G in Ĝ_{0} as a subgroups of cardinality λ). c. The Universality Spectrum of Torsion-Free Group

We may restrict ourselves in this Section to the class of reduce torsion-free groups. We shall look at some subclasses of this class. We proceed to show now that in regular λ which satisfy λ = λ^{ℵ_{0}} there is a universal reduced torsion-free group. This Theorem is an isolated point in the paper with respect to the technique, because its proof employs model theoretic notions of first order theory, elementary embedding and saturated model. These

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are available in every standard reference on model theory, like [CK]. 3.8 Theorem: if λ = λ^{ℵ_{0}} ≥ 2^{ℵ_{0}}, then there is a universal reduced torsion-free group in λ. Proof: Let T be a complete first order theory of torsion free-groups. It is enough to find a reduced group G_{T} of cardinality λ such that G_{T} ⊧ T and for every H ⊧ T, H is embedded in G_{T}; for if we have such a G for every T, the group ⨁_{T} G_{T} is of cardinality λ (there are only 2^{ℵ_{0}} complete first order theories), and is evidently universal.

Let, then, G′_{T} be a saturated model of T of cardinality λ. Let D be its maximal division subgroup, and let G_{T} =^{def} G′_{T} /D. G_{T} is isomorphic to the direct summand of D, and is therefore torsion-free and reduced. Suppose that H ⊧ T is reduced (and, clearly, torsion-free). There is an elementary embedding f : H → G′_{T}. 3.9 Claim: Imf ∩ D = 0 Proof: Suppose 0 ≠ a ∈ H and f(a) ∈ D. As f is elementary, a is divisible in H by every integer n. As H is torsion-free, the set of all divisors of a generates a divisible subgroup of H, contrary to H being reduced.

We conclude, therfore, that f̂ defined by f̂(a) = f(a) + D is an embedding of H into G_{T}. _{⌣} · · 3.8

Next we show that below the continuum there is no purely-universal reduced torsionfree group. The reason for this is trivial: there are 2^{ℵ_{0}} types (over the empty set) in this class. Therefore we do not need the club guessing machinery, and gain an extra case – the case where λ = ℵ_{1} — in comparison to Theorem 3.3. 3.10 Theorem: If λ < 2^{ℵ_{0}} then there is no purely-universal reduced torsion-free group in cardinality λ. In fact, for every reduced torsion-free group G of cardinality λ there is a rank-1 group R which is not purely embeddable in G. Proof: As λ < 2^{ℵ_{0}}, there is a characteristic (k_{1}, k_{2} ⋯), with all k_{i} finite, which is not equal

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to χ_{G}(g) for every g ∈ G (The definitions of characteristic and type are from [Fu], II, 85). Let R be a rank-1 group such that χ_{R}(1) = (k_{1}, k_{2} ⋯). As pure embeddings preserve the characteristic, R is not purely embeddable in G. _{⌣} · · 3.10

We may restrict ourselves to a subclasse of torsion-free groups which does not have many types above the empty set. This is the class of homogeneous groups (a group is homogeneous if all non zero elements in the group have the same type. See [Fu] II p.109). Here we are able to prove with the aid of club guessing, that a cardinal λ < 2^{ℵ_{0}} is not in the pure universality spectrum of the class of homogeneous groups with p-type t, if λ > ℵ_{1}. 3.11 Theorem: If λ is a regular cardinal, ℵ_{1} < λ < 2^{ℵ_{0}}, and t is a given p-type, then there is no purely universal group in λ for the class of homogeneous groups whose type is t. Proof: Let G be any homogeneous group with type t, and fix some representation G. Let C be a club guessing sequence, and for every δ ∈ S let A_{δ} be such that A_{δ} ∉ P_{δ}(G, C). Such an A_{δ} exists, as |P_{δ}(G, C)| ≤ λ < 2^{ℵ_{0}}, while there are 2^{ℵ_{0}} subsets of c_{δ}. By Theorem 2.5, there is a homogeneous group H with type t such that A_{δ} ∈ P_{δ}(H, C) for every δ ∈ S. If there were a pure embedding φ : H → G, then by Theorem 1.12, for almost every δ ∈ S, A_{δ} would be in P_{δ}(G, C). But by the choice of A_{δ} this is impossible. _{⌣} · · 3.11

We now examine cardinals λ for which λ < λ^{ℵ_{0}}. We shall prove for two classes of torsion-free groups, the class of slender groups and the class of homogeneous groups, that in certain regular λ, not only is there no universal group in the class, but also there is no torsion-free group (even outside the class) which is universal for the class. 3.12 Theorem: λ > 2^{ℵ_{0}} is a regular cardinal, and there is some μ such that μ^{+} < λ < μ^{ℵ_{0}} then there is no universal homogeneous group of p-type t in λ. In fact, there is no reduced torsion-free group G such that every homogeneous group of type t of cardinality λ is embeddable in it.

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Proof: We use the same argument as in previous theorems. We exploit Theorem 2.7 to obtain a homogeneous group of type t with a prescribed “non-embeddable” INV. _{⌣} · · 3.12

3.13 Theorem: λ > 2^{ℵ_{0}} and there is some μ such that μ^{+} < λ < μ^{ℵ_{0}} then there is no universal reduced slender group in λ. In fact, there is no reduced torsion-free group G such that every slender group of cardinality λ is embeddable in it. Proof: As before, using this time theorem 2.8. _{⌣} · · 3.13

3.14 Corollary: Under the assumptions of this theorem, there is no universal reduced torsion-free group in cardinality λ.

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Appendix

An ordinal is identified with the set of smaller ordinals. A cardinal is an ordinal on which there is no 1-1 function to a smaller ordinal. ℵ_{α} is the α-th infinite cardinal. CH is the statement 2^{ℵ_{0}} = ℵ_{1}. GCH is the statements (∀α)(2^{ℵ_{α}} = ℵ_{α+1}). The cardinality of a set X is denoted |X| and is the unique cardinal on which there is a 1-1 function unto X. The order type of a set of ordinals c, denoted by otp c, is the unique ordinal which is order isomorphic to c. The smallest infinite cardinal is ω, which is also the set of all natural numbers. By P(X) we denote the power set of X. By ^{n}X and ^{<ω}X we denote, respectively, the set of all sequences of length n of members X and the set of all finite sequences of members of X. If η is a sequence, lg η is the length of η.

An ideal over a set X is a collection of subsets of X, J ⊆ P(X) which satisfies

(1) A ∈ J and b ⊆ A ⇒ B ∈ J;

(2) A, B ∈ J ⇒ A ∪ B ∈ J. If X ∉ J, then J is a proper ideal. If J is closed under unions of lenght smaller that

some cardinal λ, then J is a λ-complete ideal. An ideal over a set is a notion of “small” sets (like measure zero sets).

An ordinal σ is a limit of a set of ordinals C if for every α < σ there is β, α < β < σ and β ∈ c. A set c of ordinals is closed if for every ordinal σ which is a limit of c, σ ∈ C. A subset c of an ordinal σ is bounded in σ if there is some α < σ such that β ∈ c ⇒ β ≤ α. If c ⊆ σ is not bounded in σ, it is unbounded in σ. An infinite cardinal λ is regular if every subset of λ which is of cardinality smaller than λ is bounded in λ.

A subset E ⊆ λ, club (of λ) if E is closed and unbounded in λ. 4.1 Fact: If λ is an uncountable regular cardinal and E_{i} ⊆ λ is a club for i < i(∗), i(∗) < λ, then ⋂ i < i(∗)E_{i} is a club. Proof: Any intersection of closed sets is closed. Suppose that α < λ is given, and we shall

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find β > α, β ∈ ⋂_{i<i(∗)} E_{i}. By induction on i < i(∗) choose α_{i}^{0} ∈ E_{i} such that α_{0}^{0} > α and i < j ⇒ α_{i}^{0} < α_{j}^{0}. This is always possible as the set {α_{j}^{0} : j < i} is bounded because of the regularity of λ, while E_{i} is unbounded. The limit of ⟨α_{i}^{0} : i < i(∗)⟩ is α_{0} and is smaller than λ. Let α_{n+1} be obtained from α_{n} in the same way α_{0} was obtained from α. The limit β of ⟨α_{n} : n < ω⟩ is a limit of each E_{i}, i < i(∗), and therefore β ∈ ⋂ i < i(∗)E_{i}. Clearly β > α. _{⌣} · · 4.1 4.2 Corollary: If λ is a regular uncountable cardinal, then the collection of all subsets X ⊆ λ with the property that λ \ X contains a club is a proper λ-complete ideal over λ. In such a case we call a set S ⊆ λ stationary if S is not in this ideal. 4.3 Fact: Suppose that λ is regular uncountable, and that λ = ⋃_{i<λ} A_{i} = ⋃_{i<λ} B_{i} where for each i < l, |A_{i}|, |B_{i}| < l, i < j < λ ⇒ A_{i} ⊆ A_{j} & B_{i} ⊆ B_{j} and whenever i is a limit ordinal, A_{i} = ⋃_{j<i} A_{j} and B_{i} = ⋃_{j<i} B_{j}. Then there is a club E ⊆ λ such that i ∈ E ⇒ A_{i} = B_{i}. Proof: : Let i_{0} < λ be given. As |A_{i_{0}}| < λ, there must be, by the regularity of λ, some i_{1} such that A_{i_{0}} ⊆ B_{i_{1}}. Symmetrically, there must be i_{2} such that B_{i_{1}} ⊆ A_{i_{2}}. Find by induction a sequence ⟨i_{n} : n < ω⟩ such that A_{i_{2n}} ⊆ B_{i_{2n+1}} and B_{i_{2n+1}} ⊆ A_{i_{2n+2}} for every n < ω. If j is the limit of ⟨i_{n} : n < ω⟩, then A_{j} = ⋃ 2nA_{i_{2n}} = ⋃_{n} B_{i_{2n+1}} = B_{j}. This shows unboundedness. Closedness is trivial. _{⌣} · · 4.3

References

[CK] C. C Chang and J. J. Keisler Model Theory, North Holland 1973.

[Fu] L. Fuchs, Infinite Abelian Groups, vols. I, II, Academic press, 1970.

[EM] P. C. Eklof and A. H. Mekler, Almost free Modules: Set theoretic methods North Holland Math. Library, 1990

[Sh-g] Saharon Shelah, Cardinal Arithmetic, to appear in OUP. [KjSh 409] M. Kojman and S. Shelah, Existence of universal models, accepted to the Journal of

Symbolic Logic

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[KjSh 447] M. Kojman and S. Shelah, The universality spectrum of stable unsuperstable theories,

accepted to Annals of Pure and Applied Logic.

[GrSh 174] R. Grossberg and S. Shelah, On universal locally finite groups, Israel J. of Math., 44, (1983), 289-302.

Math department, The Hebrew University of Jerusalem, Jerusalem 91904