PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 31–44 DOI: 10.2298/PIM150510030M

ON COMMUTATIVITY OF QUASI-MINIMAL GROUPS

Slavko Moconja

Abstract. We investigate if every quasi-minimal group is abelian, and give a positive answer for a quasi-minimal pure group having a ∅-definable partial order with uncountable chains. We also relate two properties of a complete theory in a countable language: the existence of a quasi-minimal model and the existence of a strongly regular type. As a consequence we derive the equivalence of conjectures on commutativity of quasi-minimal groups and commutativity of regular groups.

1. Introduction

In the nineties Zilber initiated the study of model theory of the complex exponential field (ℂ, +, ·, exp) and conjectured that every definable subset of ℂ is either countable or co-countable (the complement is countable); uncountable first order structures with this property are called quasi-minimal (see [9]). This conjecture is still widely open and motivates the study of model theoretical properties of quasiminimal structures, especially algebraic ones. The interesting infinitary properties are related to L_{ω_{1} ,Q} -categoricity of the complex exponential field and Schanuel’s conjecture of transcendental number theory. A complete review of this topic can be found in the first part of Zilber’s recent paper [10]. In this article we are interested in elementary first order properties of quasi-minimal structures. This direction was initiated by Itai, Tsuboi and Wakai in [1]. In particular, we will investigate quasi-minimal groups. By a group in this article we mean a first-order structure (G, ·, …) which beside the group structure may have additional operations and relations; groups with no additional structure are called pure groups.

Initially, quasi-minimality was viewed as a generalization of minimality; an infinite first order structure is minimal if any definable subset is either finite or cofinite. Minimal pure groups were classified by Reineke in 1975. In [6] he proved that every minimal group is abelian, and then by purely algebraic arguments obtained the full list. Concerning minimal fields, it is known that every algebraically closed field is a minimal structure. The converse was conjectured by Podewski in 1973 (see [5]).

2010 Mathematics Subject Classification: 03C45; 03C60; 20A15.

Key words and phrases: quasi-minimal group, strongly regular type.

Supported by the Ministry of Education, Science and Technological Development of Serbia, grant ON174018.

Communicated by Žarko Mijajlović.

Conjecture 1.1. Every minimal field is algebraically closed.

Podewski’s conjecture was confirmed by Wagner for fields of positive characteristic in [8]. Partial results in characteristic 0 were obtained by Krupinski et al. in [3], where the conjecture is reduced to fields with a proper partial ordering definable in the field language. There are natural conjectures generalizing Podewski’s conjecture and Reineke’s theorem in the quasi-minimal case.

Conjecture 1.2. Every quasi-minimal field is algebraically closed.

Conjecture 1.3. Every quasi-minimal group is abelian.

These conjectures were posed in [4] and studied recently by Gogacz and Krupinski in [2]. Gogacz and Krupinski noted that proofs of the above mentioned results on Podewski’s conjecture translate to the context of Conjecture 1.2. Concerning Conjecture 1.3 they proved that in every quasi-minimal non-abelian group all non-central elements are conjugated and that their centralizers are countable. By applying iterated HNN extensions they constructed an uncountable group with those properties, but it is still open if it is quasi-minimal. In this paper we will confirm Conjecture 1.3 in the case when there exists a ∅-definable partial order with uncountable chains. This assumption is motivated by Theorem 5.1 from [4], according to which there exist two kinds of quasi-minimal structures whose generic type is countably based (which holds in the group case): they are either symmetric or there exists a definable, over a finite parameter set, partial order with ‘large’ chains. So, we prove the following theorem.

Theorem 1.1. Every quasi-minimal pure group having a ∅-definable partial order with an uncountable chain is abelian.

The notion of regular types first arose in Shelah’s investigations of stable theories. Recently, Pillay and Tanović in [4] introduced the concept of strong regularity for global types in the general first order context (we will further include the definition of this and all other undefined notions used in the introduction). We will investigate the connection between notions of global strong regularity and quasiminimality and prove the following theorem, indicating that quasi-minimality is closely related to strong regularity.

Theorem 1.2. Assume that T is a complete theory in a countable language with infinite models.

(1) If there exists a global, countably invariant, strongly regular via x = x type, then T has a quasi-minimal model.

(2) If M ⊧ T is quasi-minimal and its generic type p is definable, then there exists a global, M-invariant, strongly regular via x = x type extending p.

Since the generic type of any quasi-minimal or strongly regular group is definable over ∅ (see Lemma 4.1 below), as an immediate corollary we obtain.

Corollary 1.1. (1) The existence of a non-abelian quasi-minimal group is equivalent with the existence of a strongly regular non-abelian group.

(2) The existence of a quasi-minimal, non algebraically closed field is equivalent with the existence of a strongly regular, non algebraically closed field.

Part (1) of the previous theorem was independently proved in unpublished notes by Haykazyan and by Tanović. Part (2) can be derived from the results in [4]. Here we will give direct proofs of both parts.

Throughout the paper we use the standard notions from model theory. Let L be a first-order language and let M be an L-structure. By ā we denote a finite tuple of elements from M and by |ā| its length. We will write ā ∈ A, where A ⊆ M, to denote that ā is a tuple of elements of A. For a subset A ⊆ M, let L_{A} be the language obtained by adding to L constant symbols naming elements of A. For an L_{A}-formulas we say that it is a formula with parameters from A. Usually, when the meaning of the model M is clear from the context, then by a formula we mean a formula with parameters from M. By a solution of ϕ(x̄) in A ⊂ M we mean a tuple ā ∈ A^{|x̄|} for which M ⊧ ϕ(ā) holds; ϕ(A) will denote the set of all solutions of ϕ(x̄) in A. A subset D ⊆ M is definable over A, or A-definable, if D = ϕ(M) holds for some formula ϕ(x) with parameters from A; D ⊆ M is definable if it is A-definable for some A ⊆ M.

A type over A ⊆ M in variables x̄ is a set of formulas whose free variables are among x̄ and whose parameters are from A, which is consistent with Th(M). By an n-type we mean a type in n free variables, usually x_{1}, …, x_{n}. If p is an n-type (over some set of parameters), we say that ā ∈ M realizes p, and write ā ⊧ p, if ā is a solution of every formula ϕ(x̄) ∈ p. The set of all realizations of p in M is denoted by p(M). We say that p is not realized in M, or that M omits p, if p(M) = ∅. An n-type over A is complete if it contains either ϕ(x̄) or ¬ϕ(x̄), for every ϕ(x̄) with parameters from A. We usually denote complete types by p, q, r, . . .. The set of all complete n-types over A is denoted by S_{n}(A). For a tuple ā ∈ M and subset A ⊆ M, by tp(ā/A) we denote the set of all formulas with parameters from A satisfied by ā; it is a complete |ā|-type over A.

For an n-type p over A and B ⊆ A, we write p_{↾B} to denote the restriction of p to the set of parameters B, i.e., the set of all L_{B}-formulas from p. A formula ϕ(x̄) (with parameters from M) is algebraic if ϕ(M) is finite. A type p ∈ S_{n}(M) is algebraic if it contains an algebraic formula; otherwise, it is non-algebraic.

A complete type p ∈ S_{1}(B) is finitely satisfiable in A ⊆ B, if every finite subtype p_{0} ⊆ p has a realization in A, equivalently if every formula ϕ(x) ∈ p has a solution in A.

Remark 1.1. If a type p ∈ S_{1}(A) is finitely satisfiable in A, then for every B ⊇ A there exists a type q ∈ S_{1}(B) such that p ⊆ q and q is finitely satisfiable in A. For the proof, consider the set of formulas

Σ(x) = {ϕ(x, b̄) | ϕ(x, ȳ) is L-formula, b̄∈ B and ϕ(A, b̄) = A}.

Obviously, Σ(x) ∪ p(x) is consistent, since p is finitely satisfiable in A. Therefore it has an extension q ∈ S_{1}(B). The type q is finitely satisfiable in A, since otherwise there is a formula φ(x) ∈ q(x) such that φ(A) = ∅. But then ¬φ(A) = A, hence ¬φ(x) ∈ Σ(x) ⊆ q(x). A contradiction.

In fact, one can easily prove that q ∈ S_{1}(B) is finitely satisfiable in A extension of p if and only if it contains Σ(x) ∪ p(x).

Let p ∈ S_{1}(M) be a type over a model, and let M ⊆ B. An extension q ∈ S_{1}(B) of p is an heir of p if for every L_{M} -formula ϕ(x, ȳ) such that ϕ(x, b̄) ∈ q for some b̄∈ B, there exists m̄ ∈ M such that ϕ(x, m̄) ∈ p.

Monster model. Assume that T is a complete theory with infinite models. If we are interested in studying the models of T of cardinality less than κ, then under some additional set-theoretical assumptions we can fix a model of T of cardinality κ, usually denoted by ℭ, with the following properties:

• every model M of T with |M| ⩽ κ is elementary embeddable in ℭ, hence we may assume that M is an elementary submodel of ℭ;

• every type with parameters from A, where A ⊆ ℭ and |A| < κ, is realized in ℭ;

• if tp(ā/A) = tp(b̄/A), where A ⊆ ℭ and |A| < κ, then there exists an automorphism f ∈ Aut_{A}(ℭ) such that f(ā) = b̄.

We say that ℭ is a monster model of T. By a small subset or model we mean a set A ⊆ ℭ or a model M ≺ ℭ such that |A|, |M| < κ. Also ⊧ ϕ(ā) means ℭ ⊧ ϕ(ā). For a complete type with parameters from ℭ we say that it is global, and usually denote such type by 𝔭. By 𝔭_{↾A} we denote the restriction of the global type 𝔭 to the set of parameters A.

If Σ(x̄) is a set of formulas with parameters from A, where A is some small subset of ℭ, and ϕ(x̄) is a formula with parameters, then by Σ(x̄) ⊢ ϕ(x̄) we mean that whenever ā ∈ ℭ satisfies all formulas in Σ(x̄), then ā also satisfies ϕ(x̄) (the set of realisation of Σ(x̄) in ℭ is a subset of the set of realisations of ϕ(x̄) in ℭ). Now compactness inside ℭ can be stated in the following form. If Σ(x̄) ⊢ ϕ(x̄), then there exists some finite subset Σ_{0}(x̄) ⊆ Σ(x̄) such that Σ_{0}(x̄) ⊢ ϕ(x̄). Indeed, by considering x̄ as constants, the theory Σ(x̄) ∪ {¬ϕ(x̄)} is not consistent with T, since otherwise it would be a type over small set of parameters, hence realized in ℭ. By compactness, for some finite Σ_{0}(x̄) ⊆ Σ(x̄), Σ_{0}(x̄) ∪ {¬ϕ(x̄)} is not consistent with T; in particular Σ_{0}(x̄) ⊢ ϕ(x̄).

A global type 𝔭 ∈ S_{n}(ℭ) is invariant over A, or A-invariant, if for every automorphism f ∈ Aut_{A}(ℭ), every formula ϕ(x̄, ȳ) with no parameters and every ā ∈ ℭ: ϕ(x̄, ā) ∈ 𝔭 iff ϕ(x̄, f(ā)) ∈ 𝔭 holds. The type 𝔭 is invariant if it is A-invariant, for some small A, and it is countably invariant if it is A-invariant, for some countable A. Note that if 𝔭 is A-invariant, then 𝔭 is B-invariant, for every B ⊇ A.

Let 𝔭 ∈ S_{n}(ℭ) be invariant and let (I, <) be a linear order. The sequence of n-tuples (ā_{i})_{i∈I} is a Morley sequence in 𝔭 over A if ā_{i} ⊧ 𝔭_{↾Aā_{<i}} , for each i ∈ I, where ā_{<i} denotes the set {ā_{j} | j < i}.

Definable types. Let M be a first order structure and p ∈ S_{n}(M). We say that the type p is definable over A ⊆ M if for every formula ϕ(x̄, ȳ) with no parameters, there exists formula d_{p} ϕ(ȳ) with parameters from A, such that

ϕ(x̄, m̄) ∈ p if and only if M ⊧ d_{p} ϕ(m̄)

holds for every m̄ ∈ M.

We say that formula d_{p} ϕ(ȳ) is a definition over A of formula ϕ(x̄, ȳ), and the correspondence d_{p} is a defining schema of p. A type p ∈ S_{n}(M) is definable if it is definable over M.

Remark 1.2. If we work in a countable language L, then every definable type is definable over some countable set. Indeed, there are only countably many formulas with no parameters, hence if p ∈ S_{n}(M) is a definable type, then the definition of each formula uses only finitely many parameters from M, hence p is definable over some countable A ⊆ M.

2. Closure operations associated to types

We start with the definition of a closure operation on a set.

Definition 2.1. Suppose that S is a non-empty set and let cl : P(S) −→ P(S). We say that cl is closure operation on S if it satisfies (for all X, Y ⊆ S):

• X ⊆ Y implies X ⊆ cl(X) ⊆ cl(Y); (Monotonicity)

• cl(X) = ⋃ {cl(X_{0}) | X_{0} ⊆ X, X_{0} is finite}; (Finite character)

• cl(cl(X)) = cl(X). (Transitivity) We say that cl is pregeometry operation if in addition it satisfies (for all a, b ∈ S and X ⊆ S):

• b ∈ cl(X ∪ {a}) ∖ cl(X) implies a ∈ cl(X ∪ {b}). (Exchange property)

If cl : P(S) −→ P(S) is an operation on S, then for any A ⊆ S one can define another operation cl^{A} : P(S) −→ P(S) on S with

cl^{A}(X) = cl(A ∪ X).

We call the operation cl^{A} the relativization of cl in A. It is easy to see that any relativization keeps monotonicity, finite character, transitivity and exchange property.

An important notion related to a closure operation cl, which we will use, is the notion of a cl-free sequence.

Definition 2.2. Suppose that cl is a closure operation on S, and (I, ⩽) is a linear order. We say that the sequence (a_{i})_{i∈I} of elements of S, is cl-free over A ⊆ S if

a_{i} ∉ cl(A ∪ {a_{j} | j < i})

holds for every i ∈ I. In particular, (a, b) is a cl-free sequence over A ⊆ S if a ∉ cl(A) and b ∉ cl(A ∪ {a}).

Assume that p ∈ S_{1}(N) is a non-algebraic type, where N is a first order structure (possibly N = ℭ). We associate to p an operation cl_{p} on P(N) defined by

cl_{p}(X) = {a ∈ N | a |̸= p_{↾X} }.

Equivalently

cl_{p}(X) = ⋃ {ϕ(N, ā) | ϕ(x, ȳ) is L-formula, ā ∈ X and ϕ(x, ā) ∉ p}.

It is not common, but we will say that a formula is p-large if it belongs to p; otherwise it is p-small. If the type p is clear from the context, we just say that a formula is large (small), instead of p-large (p-small). So, keeping in mind this convention, we see that cl_{p}(X) is the union of all sets definable by a p-small formula with parameters from X.

Remark 2.1. Let p ∈ S_{1}(N).

(1) Operation cl_{p} always satisfies monotonicity and finite character. Since p is non-algebraic, X ⊆ cl_{p}(X) immediately follows. If X ⊆ Y, then cl_{p}(X) ⊆ cl_{p}(Y) since every formula with parameters from X is also a formula with parameters from Y. For finite character, if a ∈ cl_{p}(X), then a is a solution of some small formula with parameters from X. Since this formula uses only finitely many parameters X_{0} from X, it follows that a ∈ cl_{p}(X_{0}).

(2) As we have remarked earlier, if p is a non-algebraic type, then every relativization cl_{p}^{A} satisfies Monotonicity and Finite character.

Definition 2.3. Assume that 𝔭 ∈ S_{1}(ℭ) is a global non-algebraic type. The type 𝔭 is strongly regular if there exists some small A ⊆ ℭ such that:

• 𝔭 is A-invariant;

• for every B ⊇ A and every a |̸= 𝔭_{↾B}: 𝔭_{↾B} ⊢ 𝔭_{↾Ba} holds. In that case we say that A witnesses the strong regularity of 𝔭.

Remark 2.2. (1) Pillay and Tanović in [4] define that a global, nonalgebraic type 𝔭 ∈ S_{1}(ℭ) is strongly regular via formula ϕ(x) if there exists some small A ⊆ ℭ such that:

• ϕ(x) ∈ 𝔭_{↾A} and 𝔭 is A-invariant;

• for every B ⊇ A and every a ∈ ϕ(ℭ) ∖ 𝔭_{↾B}(ℭ): 𝔭_{↾B} ⊢ 𝔭_{↾Ba} holds. Note that our definition of a strongly regular type corresponds to the definition

of a strongly regular type via x = x in the sense of Pillay and Tanović.

(2) An equivalent characterization of a strongly regular type is the following: A global, non-algebraic type 𝔭 is strongly regular, witnessed by small A, if:

• 𝔭 is A-invariant;

• cl_{𝔭}^{A} is a closure operation on ℭ. The proof of this characterisation can be found in [4].

(3) It is clear from the definition that if A witnesses strong regularity of 𝔭, then any small B ⊇ A witnesses strong regularity of 𝔭. Moreover, if 𝔭 is strongly regular, then every small subset B, such that 𝔭 is B-invariant, witnesses strong regularity of 𝔭. For the proof of this fact see [7].

Let us fix a strongly regular type 𝔭 ∈ S_{1}(ℭ), witnessed by A. We consider cl_{𝔭}^{A}-free sequences. We will freely use the following fact.

Fact 2.1. If (I, <) is a linear order, and (a_{i})_{i∈I} and (b_{i})_{i∈I} are two cl_{𝔭}^{A}-free sequences over some small B ⊇ A, then tp((a_{i})_{i∈I} /B) = tp((b_{i})_{i∈I} /B).

Note that for a strongly regular type 𝔭, the cl_{𝔭}^{A}-free sequences over A are precisely the Morley sequences in 𝔭 over A. Indeed, if (I, <) is a linear order, (a_{i})_{i∈I} is a cl_{𝔭}^{A}-free sequence over A iff a_{i} ∉ cl_{𝔭}^{A}(ā_{<i} ) iff a_{i} ⊧ 𝔭_{↾Aā_{<i}} iff (a_{i})_{i∈I} is a Morley sequence in 𝔭 over A, where ā_{<i} denotes the set {a_{j} | j < i}.

According to the dichotomy theorem for regular types from [4] we have two kinds of regular types: symmetric and asymmetric.

Symmetric kind: If cl_{𝔭}^{A} is a pregeometry operation on ℭ.

In this case every cl_{𝔭}^{A}-free sequence over A is invariant under permutations, i.e., if (a_{1}, a_{2}, …, a_{n}) is a cl_{𝔭}^{A}-free sequence over A and π is a permutation of {1, 2 …, n}, then (a_{π(1)} , a_{π(2)} , …, a_{π(n)} ) is also a cl_{𝔭}^{A}-free sequence over A. In particular, by Fact 2.1 every cl_{𝔭}^{A}-free sequence is totally indiscernible over A.

Asymmetric kind: If cl_{𝔭}^{A} is not a pregeometry operation on ℭ. In this case there exist a finite extension A_{0} of A and a A_{0}-definable partial

order on ℭ such that every cl_{𝔭}^{A}-free sequence over A_{0} is strictly increasing. Now, we turn to the notion of a quasi-minimal structure.

Definition 2.4. A first order structure M in a countable language is said to be quasi-minimal if it is uncountable and every M-definable subset of M is either countable or co-countable (its complement in M is countable).

Naturally, we say that a definable subset is small if it is countable, and that it is large if it is co-countable. Accordingly, we say that a formula is small (large) if it defines small (large) subset.

For a quasi-minimal structure M, denote by p the set of all large formulas with parameters from M; then p ∈ S_{1}(M). Indeed, p is closed for finite conjunctions, since the intersection of finitely many co-countable subsets is co-countable. For completeness note that by quasi-minimality every formula defines either a countable or a co-countable subset of M, hence either a formula or its negation belongs to p. We say that p is the generic type of M.

Remark 2.3. Let M be a quasi-minimal structure and let p be its generic type.

(1) The set cl_{p}(X) is countable for every at most countable subset X ⊆ M. Indeed, there are only countably many formulas with parameters from X, hence cl_{p}(X) is a countable union of countable sets.

(2) Since cl_{p}(X) is defined as the union of all countable X-definable subsets, it follows that f(cl_{p}(X)) = cl_{p}(f(X)) for every automorphism f ∈ Aut(M).

(3) The operation cl_{p} doesn’t need to be a closure operation on M. An example can be found in [4].

Lemma 2.1. Assume that M is a quasi-minimal structure whose generic type p is definable over countable A ⊆ M. Then cl_{p}^{A} is a closure operation on M.

Proof. As we remarked earlier, cl_{p}^{A} satisfies Monotonicity and Finite character. It remains to prove Transitivity. We have to prove that cl_{p}^{A}(cl_{p}^{A}(X)) ⊆ cl_{p}^{A}(X), since the other inclusion holds by Monotonicity.

Let c ∈ cl_{p}^{A}(cl_{p}^{A}(X)). Then M ⊧ ϕ(c, b̄), where ϕ(x, ȳ) is a formula with parameters from A, b̄= b_{1}b_{2} …b_{n} ∈ cl_{p}^{A}(X) and ϕ(x, b̄) ∉ p. For each i, 1 ⩽ i ⩽ n, b_{i} ∈ cl_{p}^{A}(X) implies that M ⊧ ψ_{i}(b_{i}, ā), where ψ_{i}(y_{i}, z̄) is a formula with parameters from A, ā ∈ X and ψ_{i}(y_{i}, ā) ∉ p. By definability of p, ϕ(x, b̄) ∉ p and ψ_{i}(y_{i}, ā) ∉ p imply M ⊧ ¬ d_{p} ϕ(b̄) and M ⊧ ¬ d_{p} ψ_{i}(ā), where d_{p}ϕ(ȳ) and d_{p}ψ_{i}(z̄) are the definitions over A of formulas ϕ(x, ȳ) and ψ_{i}(y_{i}, z̄). So c satisfies the following formula with parameters from A ∪ X

θ(x, ā) := ∃ȳ(ϕ(x, ȳ) ∧ ¬d_{p}ϕ(ȳ) ∧ ⋀_{i=1}^{n} ψ_{i}(y_{i}, ā)).

It suffices to prove that θ(x, ā) ∉ p; c ∈ cl_{p}^{A}(X) follows. Note that:

θ(M, ā) = ⋃ {ϕ(M, d̄) | d̄∈ M, M ⊧ ¬d_{p}ϕ(d̄), M ⊧ ψ_{i}(d_{i}, ā), for all 1 ⩽ i ⩽ n}.

Since M ⊧ ¬ d_{p} ψ_{i}(ā), we conclude that there are only countably many choices for d̄ such that M ⊧ ψ_{i}(d_{i}, ā) for all 1 ⩽ i ⩽ n. Therefore, the former set is countable. For any d̄ ∈ M such that M ⊧ ¬d_{p}ϕ(d̄) and M ⊧ ψ_{i}(d_{i}, ā), for all 1 ⩽ i ⩽ n, M ⊧ ¬d_{p}ϕ(d̄) implies that ϕ(x, d̄) ∉ p, hence ϕ(M, d̄) is countable. So, θ(M, ā) is a countable union of countable sets, hence is countable. Therefore, θ(x, ā) ∉ p. □

3. Strong regularity and quasi-minimality

Throughout this section we assume that T is a complete theory in a countable language, with ℭ being its monster model. In this section we prove Theorem 1.2. Part (1) is proven in Proposition 3.1 and part (2) in Proposition 3.2. We need few lemmas.

Lemma 3.1. Assume that M is a countable model of T and ā ∈ ℭ. There exists a countable model N of T such that:

• M ⊆ N, ā ∈ N;

• for every b ∈ N ∖ M, tp(b/Mā) is not finitely satisfiable in M.

Proof. Consider the following set of formulas with parameters from Mā:

Σ(x) = {ϕ(x) | ϕ(x) is L_{Mā} –formula and ϕ(M) = M} ∪ {x ≠ m | m ∈ M}.

The set Σ(x) is obviously (incomplete) type over Mā.

We claim that the following conditions are equivalent for every q ∈ S_{1}(Mā):

(1) q is finitely satisfiable in M and q is not realized in M;

(2) Σ(x) ⊆ q(x).

(1)⇒(2): Assume that q is finitely satisfiable in M, but not realized in M. Obviously {x ≠ m | m ∈ M} ⊆ q(x), since q is complete and not realized in M. Assume that ϕ(x) ∈ Σ(x) is such that ϕ(M) = M. Since q is complete, either ϕ(x) or ¬ϕ(x) belongs to q(x). But since ϕ(M) = M, we get that ¬ϕ(x) has no solution in M, so ¬ϕ(x) ∉ q(x) since q is finitely satisfiable in M. Therefore, ϕ(x) ∈ q(x).

(2)⇒(1): Assume now that Σ(x) ⊆ q(x). Since {x ≠ m | m ∈ M} ⊆ q(x), q is not realized in M. If q is not finitely satisfiable in M, then there exists a formula ϕ(x) ∈ q(x) that has no solution in M, i.e. ϕ(M) = ∅. Then ¬ϕ(M) = M, so ¬ϕ(x) ∈ Σ(x). Therefore, ¬ϕ(x) ∈ Σ(x) ∖ q(x). A contradiction.

By the claim, if a countable model N of T, such that M ⊆ N and ā ∈ N, omits Σ(x), then for every b ∈ N ∖ M, tp(b/Mā) is not finitely satisfiable in M. (Note that tp(b/Mā) is not realized in M, since this implies b ∈ M.) Thus, by Omitting Types Theorem, it suffices to show that for no consistent formula θ(x) with parameters from Mā, θ(x) ⊢ Σ(x) holds.

Toward a contradiction, assume that θ(x) is a consistent formula with parameters from Mā such that θ(x) ⊢ Σ(x). The formula θ(x) has a solution in ℭ, so choose b ∈ ℭ such that ⊧ θ(b). Since θ(x) ⊢ Σ(x), we get that Σ(x) ⊆ tp(b/Mā), and by the claim above, tp(b/Mā) is finitely satisfiable in M. In particular, θ(x) ∈ tp(b/Mā) has a solution in M. But this is not possible since θ(x) ⊢ {x ≠ m | m ∈ M}. A contradiction. □

Lemma 3.2. Assume that 𝔭 ∈ S_{1}(ℭ) is a strongly regular type, witnessed by A. Let M be a small model containing A and let a ⊧ 𝔭_{↾M}. If tp(b/Ma) is not finitely satisfiable in M, then b ⊧ 𝔭_{↾M}.

Proof. Assume that tp(b/M) is finitely satisfiable in M, then by Remark 1.1 it has a finitely satisfiable in M extension q ∈ S_{1}(Ma). Let b′ ⊧ q. Then tp(b′/M) = tp(b/M).

Toward a contradiction, assume that b |̸= 𝔭_{↾M}. Then b′|̸= 𝔭_{↾M} as well. By strong regularity of 𝔭, we get that 𝔭_{↾M} ⊢ 𝔭_{↾Mb} and 𝔭_{↾M} ⊢ 𝔭_{↾Mb′}, so a ⊧ 𝔭_{↾Mb} and a ⊧ 𝔭_{↾Mb′}, since a ⊧ 𝔭_{↾M}. But this implies that tp(b/Ma) = tp(b′/Ma), which is a contradiction since tp(b′/Ma) is finitely satisfiable in M, and tp(b/Ma) is not. □

Corollary 3.1. Assume that 𝔭 ∈ S_{1}(ℭ) is a strongly regular type, witnessed by A. Let M be a countable model of T containing A and a ⊧ 𝔭_{↾M}. There exists a countable model N of T such that: M ⊆ N, a ∈ N and 𝔭_{↾M} (N) = N ∖ M.

Proof. By Lemma 3.1 there exists a countable model N of T such that M ⊆ N, a ∈ N and for every b ∈ N ∖ M, tp(b/Ma) is not finitely satisfiable in M. Then for every b ∈ N ∖ M, b ⊧ 𝔭_{↾M} follows by Lemma 3.2. Therefore, N ∖ M ⊆ 𝔭_{↾M} (N). The other inclusion is obvious. □

Now we are ready to prove Theorem 1.2 (1).

Proposition 3.1. Assume that 𝔭 ∈ S_{1}(ℭ) is a countably invariant, strongly regular type. Then there exists a quasi-minimal model N of T.

Proof. Let A be a countable set such that 𝔭 is A-invariant. By Remark 2.2 (3), A witnesses strong regularity of 𝔭.

We build a sequence (M_{α} , a_{α})_{α<ω_{1}} such that:

• M_{α} is a countable model of T;

• M_{α} ⊆ M_{α+1} , a_{α} ∈ M_{α+1} ∖ M_{α} and a_{α} ⊧ 𝔭_{↾M_{α}} ;

• 𝔭_{↾M_{α}} (M_{α+1} ) = M_{α+1} ∖ M_{α}. We proceed by induction. For α = 0 take any countable model M_{0} of T

containing A, and take any a_{0} ⊧ 𝔭_{↾M_{0}} ; note that a_{0} ∉ M_{0}.

If α is a limit ordinal, take M_{α} = ⋃ _{β<α} M_{β} and take any a_{α} ⊧ 𝔭_{↾M_{α}} ; note that M_{α} is a countable model of T and a_{α} ∉ M_{α}.

Given (M_{α} , a_{α}) such that M_{α} is a countable model of T containing A and a_{α} ⊧ 𝔭_{↾M_{α}} , by Corollary 3.1 there is a countable model M_{α+1} such that M_{α} ⊆ M_{α+1} , a_{α} ∈ M_{α+1} and 𝔭_{↾M_{α}} (M_{α+1} ) = M_{α+1} ∖ M_{α}. Take any a_{α+1} ⊧ 𝔭_{↾M_{α+1}} ; note that a_{α+1} ∉ M_{α+1} .

It is clear that the defined sequence (M_{α} , a_{α})_{α<ω_{1}} satisfies the desired conditions.

Take N = ⋃ _{α<ω_{1}} M_{α}. Obviously, N is an uncountable (since it contains (a_{α})_{α<ω_{1}} ) model of T. Also note that 𝔭_{↾M_{α}} (N) = N ∖ M_{α}, for every α < ω_{1}. Indeed, if b ∈ N ∖ M_{α}, then b ∈ M_{β+1} , for some β ⩾ α. By construction, b ⊧ 𝔭_{↾M_{β}} , hence b ⊧ 𝔭_{↾M_{α}} , since M_{α} ⊆ M_{β}. Therefore 𝔭_{↾M_{α}} (N) ⊇ N ∖ M_{α}. The other inclusion is obvious.

We claim that N is quasi-minimal. Let ϕ(x) be a formula with parameters from N. Then ϕ(x) is a formula with parameters from M_{α}, for some α < ω_{1}. If ϕ(x) ∉ 𝔭_{↾M_{α}} , then no element of 𝔭_{↾M_{α}} (N) = N ∖ M_{α} satisfies ϕ(x). Thus ϕ(N) ⊆ M_{α}, so ϕ(N) is countable. If ϕ(x) ∈ 𝔭_{↾M_{α}} , then ¬ϕ(x) ∉ 𝔭_{↾M_{α}} , hence ¬ϕ(N) is countable and ϕ(N) is co-countable. Therefore, N is indeed a quasi-minimal model of T. □

Corollary 3.2. Assume that 𝔭 ∈ S_{1}(ℭ) is a definable, strongly regular type. Then there exists a quasi-minimal model N of T.

Proof. Since 𝔭 is definable, by Remark 1.2 it is definable over some countable A. Now it is clear that 𝔭 is A-invariant: if ϕ(x, ȳ) is an L-formula, ā ∈ ℭ and f ∈ Aut_{A}(ℭ), then

ϕ(x, ā) ∈ 𝔭 iff ⊧ d_{𝔭} ϕ(ā) iff ⊧ d_{𝔭} ϕ(f(ā)) iff ϕ(x, f(ā)) ∈ 𝔭,

where d_{𝔭} is the defining schema of 𝔭 over A.

Therefore, 𝔭 is countably invariant, hence by Proposition 3.1, T has a quasiminimal model. □

In the following proposition we prove Theorem 1.2 (2).

Proposition 3.2. Assume that M is a quasi-minimal model whose generic type p is definable. Then the global heir 𝔭 of p is strongly regular.

Proof. Since p is definable, by Remark 1.2 it is definable over some countable A ⊆ M. Let ℭ be a monster model of a theory T = Th(M), and let d_{p} be the defining schema of the type p over A. The global heir of p is defined with:

𝔭 = {ϕ(x, ā) | ϕ(x, ȳ) is L-formula, ā ∈ ℭ and ⊧ d_{p} ϕ(ā)}.

We claim that A witnesses the regularity of 𝔭. First, note that 𝔭 is A-invariant. Assume that ϕ(x, ā) ∈ 𝔭 and f ∈ Aut_{A}(ℭ). Then ⊧ d_{p} ϕ(ā) holds, hence also ⊧ d_{p} ϕ(f(ā)) holds, since d_{p} ϕ(ȳ) is a formula with parameters from A. Therefore ϕ(x, f(ā)) ∈ 𝔭. It remains to prove that cl_{𝔭}^{A} is a closure operation on ℭ.

Note that the corresponding cl_{𝔭}^{A} operation on ℭ is defined by

cl_{𝔭}^{A}(X) = ⋃ {ϕ(ℭ, ā) | ϕ(x, ȳ) is L-formula, ā ∈ A ∪ X and ϕ(x, ā) ∉ 𝔭}

= ⋃ {ϕ(ℭ, ā) | ϕ(x, ȳ) is L-formula, ā ∈ A ∪ X and ⊧ ¬ d_{p} ϕ(ā)}.

Assume on the contrary that cl_{𝔭}^{A} is not a closure operation on ℭ. Then Transitivity condition fails, so we can choose X ⊆ ℭ and c ∈ ℭ such that c ∈ cl_{𝔭}^{A}(cl_{𝔭}^{A}(X)) ∖ cl_{𝔭}^{A}(X). Since c ∈ cl_{𝔭}^{A}(cl_{𝔭}^{A}(X)), there exists a formula ϕ(x, ȳ) with parameters from A and b̄= b_{1}b_{2} …b_{n} ∈ cl_{𝔭}^{A}(X) such that ϕ(x, b̄) ∉ 𝔭 and ⊧ ϕ(c, b̄). For each i, 1 ⩽ i ⩽ n, b_{i} ∈ cl_{𝔭}^{A}(X) implies that there exists a formula ψ_{i}(y_{i}, z̄) with parameters from A and ā ∈ X such that ψ_{i}(y_{i}, ā) ∉ 𝔭 and ⊧ ψ_{i}(b_{i}, ā). By the definition of 𝔭, ϕ(x, b̄) ∉ 𝔭 and ψ_{i}(y_{i}, ā) ∉ 𝔭 imply ⊧ ¬ d_{p} ϕ(b̄) and ⊧ ¬ d_{p} ψ_{i}(ā).

Therefore, b̄ witnesses the existential quantifier that c satisfies

θ(x, ā) := ∃ȳ[ϕ(x, ȳ) ∧ ¬ d_{p} ϕ(ȳ) ∧ ⋀_{i=1}^{n} ψ_{i}(y_{i}, ā) ∧ ⋀_{i=1}^{n} ¬ d_{p} ψ_{i}(ā)],

where θ(x, z̄) is a formula with parameters from A. Since c ∉ cl_{𝔭}^{A}(X) and ā ∈ X we conclude that θ(x, ā) ∈ 𝔭, i.e., ⊧ d_{p} θ(ā), and accordingly: ⊧ ∃z̄d_{p} θ(z̄) holds.

Since M ≺ ℭ and ∃z̄d_{p} θ(z̄) is a formula with parameters from A ⊆ M, M ⊧ ∃z̄d_{p} θ(z̄) also holds, i.e., there exists ā′ ∈ M such that M ⊧ d_{p} θ(ā′). This implies that θ(x, ā′) ∈ p, i.e. θ(M, ā′) is co-countable. Since cl_{p}^{A}(ā′) is countable, we can choose c′ ∈ θ(M, ā′) ∖ cl_{p}^{A}(ā′). Now M ⊧ θ(c′, ā′) implies that there exists b̄′∈ M such that

M ⊧ ϕ(c′, b̄′) ∧ ¬ d_{p} ϕ(b̄′) ∧ ⋀_{i=1}^{n} ψ_{i}(b′_{i}, ā′) ∧ ⋀_{i=1}^{n} ¬ d_{p} ψ_{i}(ā′).

The last two conjuncts imply that b̄′ ∈ cl_{p}^{A}(ā′), and then the first two say that c′ ∈ cl_{p}^{A}(cl_{p}^{A}(ā′)). Therefore c′ ∈ cl_{p}^{A}(cl_{p}^{A}(ā′)) ∖ cl_{p}^{A}(ā′). This is a contradiction, since by Lemma 2.1, cl_{p}^{A} is a closure operation on M. □

4. Around quasi-minimal group

In this section we prove Theorem 1.1.

According to the general definition of a quasi-minimal structure, a group G in a countable language is quasi-minimal if it is uncountable and every G-definable subset of G is either countable or co-countable (its complement in G is countable).

Let p be the generic type of G, i.e. the type consisting of all large formulas with parameters from G. It turns out that the type p is definable over ∅. This fact is well known (see [2, 3, 4]), but for the sake of completeness we formulate and prove it in the following lemma.

Lemma 4.1. The generic type p of a quasi-minimal group G is definable over ∅.

Proof. First we prove that for any co-countable subset S ⊆ G there exists a ∈ G such that S ∪aS = G. Since S is co-countable, then also S^{−1} is co-countable, as well as gS^{−1} , for each g ∈ G. Consider the family {gS^{−1} | g ∉ S}. Since it is a countable family of co-countable subsets, it has co-countable intersection, so choose a in its intersection. But then for each g ∉ S we have a ∈ gS^{−1} , hence g ∈ aS. Therefore G = S ∪ aS.

Also note that if G = S ∪ aS holds for a subset S ⊆ G, then S is uncountable (since G is uncountable). Therefore, if in addition S is definable, then by quasiminimality S is co-countable.

Suppose now that ϕ(x, ȳ) is a formula with no parameters. Note that for any ḡ ∈ G, ϕ(x, ḡ) ∈ p iff ϕ(G, ḡ) is co-countable iff G = ϕ(G, ḡ) ∪ a ϕ(G, ḡ) for some a ∈ G. Therefore, ϕ(x, ḡ) ∈ p iff G ⊧ d_{p} ϕ(ḡ), where d_{p} ϕ(ȳ) is the formula

∃z ∀x [ϕ(x, ȳ) ∨ ∃x′(ϕ(x′, ȳ) ∧ x = zx′)].

Note that d_{p} ϕ(ȳ) has no parameters, hence it is the definition of ϕ(x, ȳ) over ∅. □

By Lemma 2.1 we have an immediate corollary.

Corollary 4.1. Let G be a quasi-minimal group whose generic type is p. Then cl_{p} = cl_{p}^{∅} is a closure operation on G.

In order to prove Theorem 1.1 we first consider some properties that hold if a non-abelian quasi-minimal group exists. The same properties hold in the minimal case and some of them are proven in [6]. The proof of Lemma 4.2 in the quasiminimal case can also be found in [2]. In Lemma 4.2 and Corollaries 4.2 and 4.3 we deal under the following assumption.

Assumption 4.1. Assume that G is a non-abelian quasi-minimal group.

Lemma 4.2.

(i) Every definable proper subgroup of G is countable.

(ii) The center Z(G) is countable.

(iii) For every a ∉ Z(G), the centralizer C_{G}(a) is countable and the conjuga-

tion class a^{G} is co-countable. (iv) For every a ∉ Z(G), G = Z(G) ∪ a^{G} holds.

Proof. Assume that H is a definable proper subgroup of G. Assume that H is not countable, hence by quasi-minimality it is co-countable. Choose a ∉ H. Then H and aH are disjoint co-countable sets of G. This contradiction proves (i). For (ii) note that Z(G) is definable proper subgroup of G, since G is non-abelian. Therefore, (ii) follows from (i). Similarly, for (iii) note that C_{G}(a) is definable proper subgroup of G, since a ∉ Z(G). Hence, the countability of C_{G}(a) follows from (i) as well. Since |a^{G}| = |G : C_{G}(a)| holds by the orbit-stabilizer theorem, we get that a^{G} is uncountable, hence by quasi-minimality it is co-countable.

(iv) Take a, b ∉ Z(G). By (iii) a^{G} and b^{G} are co-countable, hence they meet. Therefore a^{G} = b^{G}, and we get that G = Z(G) ∪ a^{G}. □

Corollary 4.2. For every a, b ∉ Z(G), the set {x ∈ G | a^{x} = b} is countable.

Proof. By Lemma 4.2 (iv) there exists g ∈ G such that b = a^{g}. Then b = a^{x} iff a^{g} = a^{x} iff a^{xg^{−1}} = a iff xg^{−1} ∈ C_{G}(a) iff x ∈ C_{G}(a)g. Therefore, there exists a bijection between {x ∈ G | a^{x} = b} and C_{G}(a), which is countable by Lemma 4.2 (iii). □

If in addition we assume that G is a pure group, then we get the following corollary.

Corollary 4.3. We have cl_{p}(∅) = Z(G).

Proof. Obviously, Z(G) ⊆ cl_{p}(∅), since by Lemma 4.2(ii), Z(G) is countable and also ∅-definable. Toward a contradiction, assume that Z(G) ⊊ cl_{p}(∅) and choose a ∈ cl_{p}(∅) ∖ Z(G). Then there exists ∅-definable countable subset D ⊆ G containing a. But then for every g ∈ G, a^{g} ∈ D^{g}, and D^{g} = D since D is ∅-definable and conjugation is an automorphism of a pure group. Therefore, a^{G} ⊆ D, which is a contradiction since by Lemma 4.2(iii), a^{G} is co-countable. □

Now, till the end of the section we deal under a stronger assumption.

Assumption 4.2. Assume that G is non-abelian quasi-minimal pure group with ∅-definable partial order ⩽ with an uncountable chain.

Lemma 4.3. Exactly one of the following holds:

(1) a < x defines a co-countable subset of G, for every a ∉ cl_{p}(∅);

(2) x < a defines a co-countable subset of G, for every a ∉ cl_{p}(∅).

Proof. Since cl_{p}(∅) is countable and there exists an uncountable chain C, we can choose an element a_{0} ∈ C ∖ cl_{p}(∅). Then at least one of a_{0} < x and x < a_{0} defines an uncountable subset, hence by quasi-minimality exactly one of a_{0} < x and x < a_{0} defines a co-countable subset of G.

Assume that a_{0} < x defines a co-countable subset of G and take any a ∉ cl_{p}(∅). By Corollary 4.3, cl_{p}(∅) = Z(G), so by Lemma 4.2(iv) we can choose an element g ∈ G such that a_{0}^{g} = a. Since ⩽ is 0-definable and the conjugation is an automorphism of a pure group, we get that a < x defines a co-countable subset of G as well.

In a similar way we can prove that x < a defines a co-countable subset of G, for every a ∉ cl_{p}(∅), assuming that x < a_{0} defines a co-countable subset of G. □

Lemma 4.4. For every a, b ∉ cl_{p}(∅), cl_{p}(a) ⊆ cl_{p}(b) or cl_{p}(b) ⊆ cl_{p}(a).

Proof. By Lemma 4.3, without loss of generality, assume that c < x defines a co-countable subset of G, for every c ∉ cl_{p}(∅). Then x < c and c ≮ x define countable subsets of G.

If a < b, since x < b defines a countable set, we get that a ∈ cl_{p}(b), and cl_{p}(a) ⊆ cl_{p}(b) follows by Transitivity. If a ≮ b, since a ≮ x defines a countable set, we get b ∈ cl_{p}(a), and cl_{p}(b) ⊆ cl_{p}(a) follows by Transitivity. □

Proof of Theorem 1.1. Let (a, b) be a cl_{p}-free sequence over ∅. Since b ∉ cl_{p}(a), by Lemma 4.4, cl_{p}(a) ⊊ cl_{p}(b) holds, so cl_{p}(a, b) = cl_{p}(b). By Lemma 4.2(iv) choose g ∈ G such that b = a^{g}. Since g ∈ {x ∈ G | a^{x} = b}, and since this set is countable by Corollary 4.2, we get that g ∈ cl_{p}(a, b) = cl_{p}(b), hence cl_{p}(g) ⊆ cl_{p}(b). On the other hand, obviously b ∈ cl_{p}(a, g), and since b ∉ cl_{p}(a), we conclude that g ∉ cl_{p}(a). Hence, by Lemma 4.4, cl_{p}(a) ⊆ cl_{p}(g), and finally b ∈ cl_{p}(a, g) = cl_{p}(g). Therefore, cl_{p}(b) = cl_{p}(g).

Since cl_{p}(a) ⊊ cl_{p}(b) = cl_{p}(g) and conjugation is an automorphism of a pure group, by Remark 2.3 (2) we have that cl_{p}(a^{g}) ⊊ cl_{p}(g^{g}), i.e., cl_{p}(b) ⊊ cl_{p}(g). A contradiction. □

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Faculty of Mathematics (Received 05 05 2015) University of Belgrade Belgrade Serbia

   slavko@matf.bg.ac.rs