Finitely axiomatizable strongly minimal groups

Thomas Blossier and Elisabeth Bouscaren

February 28, 2007

Abstract

We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasi-endomorphisms of G must be an infinite finitely presented ring.

Questions about finite axiomatizability of first order theories are nearly as old as model theory itself and seem at first glance to have a fairly syntactical flavor. But it was in order to show that totally categorical theories cannot be finitely axiomatized that, in the early eighties, Boris Zilber started developing what is now known as “Geometric stability theory”. Indeed, as is often the case, in order to answer such a question, one needs to develop a fine analysis of the structure of models in the class involved and to understand exactly how each model is constructed.

The easiest way to force a structure to be infinite by one first order sentence is to impose an ordering without end points, or a dense ordering, thus making the structure unstable. It was hence rather natural to wonder about theories at the other extremity of the stability spectrum, and in the early 60’s the question was posed whether there existed finitely axiomatizable totally categorical theories or simply uncountably categorical theories [21], [16].

Each model of a totally categorical theory is prime over a strongly minimal set. It is not too difficult to see that a totally categorical strongly minimal set cannot be finitely axiomatizable ([14]). Much more complicated, the proof of the non finite axiomatizability for the whole class goes through a characterization of the geometries associated to totally categorical strongly minimal sets (locally modular and locally finite) and then an analysis of how any model is “built” around the strongly minimal set ([22], [23] and [6] where the result is proved for all ω-stable ω-categorical theories).

Around the same time as Zilber’s negative answer for the totally categorical case, Peretjat’kin produced an example of a finitely axiomatized ℵ_{1}-categorical theory [18]. This example was in the following years simplified by Baisalov (see [9, 12.2, Example 5]). This final example has Morley Rank equal to 2, thus still leaving open the question of the existence of a finitely axiomatizable strongly minimal set (Morley rank and degree equal to 1). Furthermore all the known examples of finitely axiomatizable ℵ_{1}-categorical theories are rather similar and constructed around a strongly minimal set with trivial pregeometry, also leaving open the question of the existence of a finitely axiomatizable ℵ_{1}-categorical theory with non trivial pregeometry.

In 1994, Hrushovski ([10]) showed that any finitely axiomatizable ℵ_{1}-categorical theory must have locally modular pregeometry, thus reducing the remaining open questions to two very different cases:

– the existence of a finitely axiomatizable trivial strongly minimal set

– the existence of any finitely axiomatizable ℵ_{1}-categorical theory which contains a locally modular strongly minimal group.

The canonical example of a strongly minimal locally modular non trivial theory is that of infinite K-vector spaces, for a fixed division ring K. It is open whether there exists any finitely axiomatizable complete theory of R-modules, for R any ring, but it is very easy to check that if K is an infinite division ring which is finitely presented as a ring, then the theory of K-vector spaces can be finitely axiomatized. Unfortunately, the existence of such a division ring is open (see section 3). Conversely, it was originally shown by Paljutin ([17]), in a paper where he characterizes finitely axiomatizable uncountably categorical quasi-varieties, that, if the theory of infinite K-vector spaces is finitely axiomatizable, then K is finitely presented as a ring (see section 3.2).

In the paper cited above, Hrushovski conjectures that, more generally, a finitely axiomatizable ℵ_{1}-categorical non trivial theory exists if and only if such an infinite finitely presented division ring exists. Any ℵ_{1}-categorical non trivial locally modular theory must contain a locally modular strongly minimal group G, and the geometry associated to such a group is that of infinite K-vector spaces, where K is the division ring of quasi-endomorphisms of G (see section 2 for the definitions). The precise conjecture in [10] is that, in any finitely axiomatizable ℵ_{1}-categorical non trivial theory, the associated division ring of quasi-endomorphisms is infinite and finitely presented as a ring.

One should remark that although every ℵ_{1}-categorical non trivial locally modular theory must contain a definable strongly minimal group, one cannot use general arguments to transfer down the finite axiomatizability to the strongly minimal group. We will see in section 3.1 some general assumptions under which finite axiomatizability can be transferred (bi-interpretability, definable finite partition). But, it is not even true in general that, if M is finitely axiomatizable and contained in the algebraic closure of a strongly minimal set D (M is then said to be almost strongly minimal), the strongly minimal set D, with the induced structure from M, must be finitely axiomatizable. In the finitely axiomatized ℵ_{1}-categorical Morley rank 2 theory which was mentioned above, for example, the whole structure M is contained in the algebraic closure of a strongly minimal subset D ⊂ M × M (the diagonal), whose induced structure is that of the integers with the successor function, which is not finitely axiomatizable.

One last remark, in past years, work around strongly minimal finitely axiomatizable trivial sets has also centered around a conjecture relating their existence to the existence of an infinite group with specific properties (see section 3 for some further details).

In this paper we show that Hrushovski’s conjecture holds for strongly minimal groups, and more generally for Morley Rank one groups: If G is a finitely axiomatizable strongly minimal group, then the division ring of quasi-endomorphisms of G must be infinite and finitely presented (Theorem 3.19).

By Hrushovski’s result, we know that such a group must be locally modular. This enables us to reduce to the case when G is a strongly minimal Abelian structure. Then we show (Proposition 3.16) that if G is a finitely axiomatizable strongly minimal abelian structure, the division ring K of quasi-endomorphisms of G must be infinite and that the theory of K-vector spaces must also be finitely axiomatizable (Lemma 3.15).

We begin in section 1 by recalling or proving some general facts about abelian structures, under the precise form they will be needed later. In particular, we describe, in section 1.3, the theory which will end up being both finitely axiomatizable and interdefinable with the theory of K-vector spaces. In section 2 we recall the basic facts about the ring of quasi-endomorphisms of a locally modular strongly minimal group and we look at strongly minimal abelian structures. In section 3, we consider the question of finite axiomatization. We begin by a somewhat technical section (3.1) where we give precise definitions of finite axiomatizability in the case of infinite languages and we show how this notion transfers when changing languages or structures. In order to be as selfcontained as possible on the subject of finite axiomatizability, in section 3.2, we recall very precisely the two classical examples (regular group actions and vector spaces). In the next section (section 3.3), we prove the main theorem, that if a strongly minimal abelian structure is finitely axiomatizable, then its division ring of quasi-endomorphisms, K, must be infinite and the theory of K-vector spaces must be finitely axiomatizable. Finally in the last section (3.4) we conclude for strongly minimal groups and more generally for groups of Morley Rank one.

We would like to thank the many people with whom we have had very helpful discussions since we started getting interested in questions of finite axiomatizability, in particular, Ehud Hrushovski, Alexandre Ivanov, David Lippel, Dugald MacPherson, Mike Prest and Gabriel Sabbagh.

1 Abelian structures

1.1 Axiomatization and quantifier elimination

In this first section, we recall the precise statements about axiomatization and ppelimination of quantifiers for abelian structures

We define an abelian structure 𝔾, to be a commutative group

𝔾 =< G, +, −, 0, (H_{i})_{i∈I} >,

where each H_{i} is a subgroup of some G^{n_{i}}. We denote by L_{0} the language {+, −, 0, (H_{i})_{i∈I}}. We are going to consider expansions of abelian structures by constants and we will denote by L_{c} the language of an expansion of 𝔾 by some constants in a subset C, i.e. L_{c} = L_{0} ∪ {c : c ∈ C}.

Recall that the set of positive primitive formulas is the closure of the atomic formulas by conjunction and existential quantifiers.

It has been well-known for years that in a complete theory of modules, every formula is equivalent to a Boolean combination of pp-formulas and that a complete theory of modules is axiomatized by so-called invariant statements describing the index of pairs of positive primitive definable subgroups ([3], [20]).

The similar result for abelian structures has also been known for a long time (abelian structures were originally introduced by E. Fisher in [7]) but was never published in any “official” form until it appeared as a special case in the general treatment of theories given by cosets in [8]. As we are dealing with questions of finite axiomatization, it is important for us to be extremely precise about the form of the axioms and the language we are working in. For this reason we will recall briefly the precise definitions we need and state, mostly without proofs, the results under the exact form we require. Some similar considerations appear also in [5].

Lemma 1.1. Let ϕ(x_{1}, …, x_{n}) be a consistant positive primitive formula in L_{0} in n variables (n ≥ 1) and without parameters. Then {a ∈ G^{n} : 𝔾 ⊧ ϕ(a)} is a subgroup of G^{n}.

Note that the set of pp-definable subgroups in 𝔾 corresponds to the closure of the groups (H_{i} : i ∈ I), the trivial groups ({0} and G), the diagonal of G^{2} and the graph of the addition by, cartesian product, permutation of coordinates, intersection and projection.

Lemma 1.2. Let ϕ(x̄, ¯0) be a pp-formula from L_{0} without parameters, which defines a subgroup in G^{n}. Let d̄ be a tuple from G. Then ϕ(x̄, d̄) is empty or is a coset of the pp-definable subgroup defined by ϕ(x̄, ¯0).

A pp-formula in the language L_{c} is equivalent to ϕ(x̄, c̄) where ϕ is a pp-formula from L_{0} and c̄ is a tuple of constants. In particular a subgroup of G^{n} which is definable by a pp-formula from L_{c} is in fact already pp-definable in L_{0} without parameters.

Let T (𝔾) be the following set of sentences from L_{0} :

– G is a commutative group,

– for each original predicate H_{i} from L_{0}, H_{i} is a subgroup of G^{n_{i}},

– the equivalence sentences: all sentences of the form ∀x̄ (ϕ(x̄) ↔ ψ(x̄)), for ϕ and ψ pp-formula which define the same subgroup of G^{n} (note that these sentences give the following relations between pp-definable subgroups : inclusion, intersection, projection and equality up to a permutation of variables),

– the dimension sentences: for each pair H ⊂ H′ of pp-definable subgroups of G, such that the index of H in H′ is equal to n, the sentence “[H′ : H] = n”; for each pair H ⊂ H′ of pp-definable subgroups of G, such that the index of H in H′ is infinite in G, the infinite scheme of sentences “[H′ : H] ≥ k” , for every k ≥ 1.

Fact 1.3. The theory T (𝔾) is complete in the language L_{0} and admits quantifier elimination to the pp-formulas, that is, every formula is equivalent modulo T (𝔾) to a Boolean combination of pp-formulas.

Note that it follows easily that every abelian structure is stable.

Corollary 1.4. The theory of 𝔾 in the language L_{c} is given by T (𝔾) together with the pp-type of the constants (i.e. for each pp-definable group H, we have to describe the H-congruences on the set of constants).

Corollary 1.5. Let a ∈ G and B ⊂ G, then a is algebraic on B in the L_{c}-structure 𝔾 if and only if a is in a B-definable coset of some finite pp-definable subgroup of G.

Proof. By elimination to pp-formulas, the type of a over B is given by the set X of B-definable cosets of pp-definable subgroups to which a belongs, and the set Y of Bdefinable cosets of pp-definable subgroups to which a does not belong. Note that X is closed under finite intersections. By compactness, as a is algebraic over B, there is some coset A in X and some cosets B_{1}, B_{2}, ..., B_{n} from Y such that A \ (B_{1} ∪ … ∪ B_{n}) is finite non empty. We can suppose that each B_{i} is contained in A, by taking its intersection with A. Then A = a + H = {a_{1}} ∪ … ∪ {a_{m}} ∪ (d_{1} + H_{1}) ∪ … ∪ (d_{n} + H_{n}), where {a_{i}} is considered as a coset of the trivial group, d_{i} + H_{i} = B_{i} and H_{i} ⊂ H. By Neumann’s Lemma, if some coset a + H is covered by a finite number of cosets, then it is covered by those cosets which correspond to subgroups of finite index in H. It follows that A itself is finite.

We finish with a remark that will be very useful in the sequel:

Proposition 1.6. Let 𝔾 be an abelian structure in L_{c}. Let H ⊂ G^{n} be any definable connected subgroup (with parameters). Then H is pp-definable. In particular, H is definable over ∅.

Proof. Let H be a connected definable subgroup of G^{n}. By pp-elimination, there is a set A such that H is equivalent to a Boolean combination of pp-formulas with parameters in A. We can suppose that the unique generic type of H, q, is defined and stationary over A. As q is a complete type over A, we can suppose that there are ā_{0}, ā_{1}, …, ā_{m} ∈ A and pp-definable (over ∅) subgroups ϕ_{0}, ϕ_{1}, …, ϕ_{m} of G^{n} such that:

1. for each i > 0, ϕ_{i} is a subgroup of ϕ_{0},

2. (ā_{0} + ϕ_{0}) \ (ā_{1} + ϕ_{1} ∪ ... ∪ ā_{m} + ϕ_{m}) ⊆ H,

3. every generic of H over A is in (ā_{0} + ϕ_{0}) \ (ā_{1} + ϕ_{1} ∪ ... ∪ ā_{m} + ϕ_{m}),

4. for each i > 0, each ϕ_{i} is of infinite index in ϕ_{0}: indeed if ϕ_{i} has finite index in ϕ_{0}, by enlarging A if necessary, we can suppose that ā_{0} + ϕ_{0} = ⋃_{1≤j≤k} ē_{j} + ϕ_{i}, with ē_{j} ∈ A, and replace ā_{0} + ϕ_{0} with one of the ē_{j} + ϕ_{i}.

It follows that H = ϕ_{0}: the difference of two generics of H (over A) is in ϕ_{0}, so H ⊂ ϕ_{0}. Conversely, let h be a generic of H over A. Take a generic x of ϕ_{0} over Ah. Since for each i > 0, ϕ_{i} is a subgroup of ϕ_{0} of infinite index, x ∉ (ā_{i} − h) + ϕ_{i}. Thus x + h ∈ (ā_{0} + ϕ_{0}) \ (ā_{1} + ϕ_{1} ∪ ... ∪ ā_{m} + ϕ_{m}) and x ∈ H.

1.2 Direct sums

The following direct sum construction plays an essential role in the paper.

Let 𝕊 be an abelian structure in the language L_{0}, and let 𝔾 be an abelian structure in the language L_{c}. Let 𝔾_{S} := 𝔾 ⊕ 𝕊, be the L_{c}-structure with universe the group G_{S} = G ⊕ S and with the obvious interpretation of the symbols in the language: each constant c is interpreted by (c, 0); for H of arity n, we interpret H in G_{S} by H(G)⊕H(S) where H(G) := {(h, 0) ∈ G_{S}^{n} : 𝔾 ⊧ H(h)} and H(S) := {(0, h) ∈ G_{S}^{n} : 𝕊 ⊧ H(h)}. If g = (g_{1}, …, g_{n}) ∈ G^{n} and s = (s_{1}, …, s_{n}), we will use both notations g + s or (g, s) to denote the element (g_{1} + s_{1}, …, g_{n} + s_{n}) ∈ G_{S}^{n}.

Lemma 1.7. For each pp-formula ϕ(x) (x = (x_{1}, …, x_{n})), from L_{0}, ϕ(G_{S}) = ϕ(G) ⊕ ϕ(S), i.e., for h_{1} ∈ G, and h_{2} ∈ G_{S}, 𝔾_{S} ⊧ ϕ(h_{1} + h_{2}) if and only if 𝔾 ⊧ ϕ(h_{1}) and 𝕊 ⊧ ϕ(h_{2}).

Proof. By induction on pp-formulas. For atomic formulas, it follows from the way 𝔾_{S} is defined as being the direct sum of 𝔾 and 𝕊 as L_{0}-structures. For a conjunction of two ppformulas, it follows easily from the fact that G_{S} is the direct sum of G and S as groups. There remains to check the case of a projection. Let ϕ(x_{1}, ..., x_{n}) be a pp-formula such that ϕ(G_{S}) = ϕ(G) ⊕ ϕ(S) and consider the pp-formula ψ := ∃x_{1}ϕ. We have trivially that ψ(G) ⊕ ψ(S) ⊂ ψ(G_{S}). Let (a_{2}, ..., a_{n}) ∈ ψ(G_{S}). Then there is a_{1} ∈ G_{S} such that (a_{1}, ..., a_{n}) ∈ ϕ(G_{S}) = ϕ(G) ⊕ ϕ(S). So (a_{1}, ..., a_{n}) = (b_{1}, ..., b_{n}) + (c_{1}, ..., c_{n}), with (b_{1}, ..., b_{n}) ∈ ϕ(G) and (c_{1}, .., c_{n}) ∈ ϕ(S). But then (a_{2}, ..., a_{n}) = (b_{2}, ..., b_{n}) + (c_{2}, ..., c_{n}) with (b_{2}, ..., b_{n}) ∈ ψ(G) and (c_{2}, .., c_{n}) ∈ ψ(S).

Remark. In particular, if ϕ(x̄, ȳ) is a pp-formula in L_{0} and c̄ = c_{1}, …, c_{n} are some constants from C. Then 𝔾_{S} ⊧ ϕ(g + s, c) if and only if 𝔾 ⊧ ϕ(g, c) and 𝕊 ⊧ ϕ(s, 0).

1.3 The totally transcendental case

We consider an abelian structure 𝔾 with constants, in the language L_{c}, and we denote its theory by 𝕋.

Recall that in a totally transcendental group, there is no infinite strictly decreasing sequence of definable subgroups. It follows easily that the same is true for cosets:

Lemma 1.8. If M is a totally transcendental group then every infinite intersection of definable cosets in M^{n} is equivalent to a finite intersection.

The following lemmas (1.9 to 1.12) are well known for the case of modules (in the usual language for modules, see [20]) and mostly folklore for abelian structures. The proofs are similar to the ones in the case of modules.

Definition. Let 𝕄 be a model of 𝕋. We say that 𝕄 is pure injective if every set of pp-formulas with parameters in M (with possibly infinitely many variables) which is finitely realized in M is realized in M.

Lemma 1.9. If 𝕋 is totally transcendental then every model of 𝕋 is pure injective.

Proposition 1.10. If 𝔾_{0} is an elementary substructure of 𝔾 and is pure injective, then there exists f, an L_{c}-homomorphism from G to G_{0}, such that f is the identity on G_{0}.

Corollary 1.11. If 𝔾_{0} is an elementary substructure of 𝔾, then G = G_{0} ⊕ G_{1} and for each pp-definable subgroup H ⊂ G^{n} in L_{0}, H = H ∩ (G_{0}^{n}) ⊕ H ∩ (G_{1}^{n}).

Proof. Let G_{1} = ker f where f is given by the previous proposition. Then g = f(g) + (g − f(g)) and g − f(g) ∈ Kerf for each g ∈ G: indeed f(g − f(g)) = f(g) − f(f(g)) = f(g) − f(g) = 0. We have the same property for cartesian product. So we can suppose that H ⊂ G to simplify the notation. Since H is a pp-definable subgroup and f is an L_{0}-homomorphism, f(H) ⊂ H ∩G_{0} and so f(H) = H ∩G_{0}. Thus H = H ∩G_{0} ⊕H ∩G_{1}: for each h ∈ H, f(h) ∈ H ∩ G_{0} and (h − f(h)) ∈ ker f ∩ H.

Let G = G_{0} ⊕ G_{1}, as above, where 𝔾_{0} is an elementary substructure of 𝔾, and G_{1} = ker f, f given by Prop. 1.10. Let 𝔾_{1} be the following abelian structure on G_{1} in the language L_{0} :

𝔾_{1} =< G_{1}, +, −, 0, (H_{i} ∩ G_{1}^{n_{i}})_{i∈I} > .

Then 𝔾 = 𝔾_{0} ⊕ 𝔾_{1} as abelian structures (see section 1.2) and by lemma 1.7:

Lemma 1.12. For every subgroup H of G^{n} definable by a pp-formula ϕ in 𝔾, the group ϕ(G_{1}) := {a ∈ G_{1}^{n} : 𝔾_{1} ⊧ ϕ(a)} is equal to H ∩ G_{1}^{n}.

Let 𝕋_{1}(𝔾) be the following modification of the axioms T (𝔾) :

- the axioms for abelian groups,

- for each original predicate H from L_{0}, “H is a subgroup”,

- the equivalence sentences from T (𝔾),

- “[ϕ : ψ] is infinite” for every pair of pp-formulas such that in 𝔾, ψ(G) ⊂ ϕ(G) ⊂ G and [ϕ(G) : ψ(G)] is infinite,

- [ϕ : ψ] = 1 for every pair of pp-formulas such that in 𝔾, ψ(G) ⊂ ϕ(G) ⊂ G and [ϕ(G) : ψ(G)] is finite.

Proposition 1.13. If 𝔾 is |G_{0}|^{+}-saturated, 𝕋_{1}(𝔾) axiomatizes the complete theory of the abelian structure 𝔾_{1}, that is 𝕋_{1}(𝔾) ⊢ T (𝔾_{1}).

Proof. We show first that 𝔾_{1} is a model of 𝕋_{1} := 𝕋_{1}(𝔾). By Lemma 1.12, 𝔾_{1} satisfies the equivalence sentences from T (𝔾). Let K ⊂ H be a pair of pp-definable subgroups of G. If [H : K] is infinite then by |G_{0}|^{+}-saturation of 𝔾, [H : K] > |G_{0}|. Thus [H ∩G_{1} : K ∩G_{1}] is infinite. If [H : K] is equal to k, since G_{0} is an elementary substructure, there are a_{1}, ..., a_{k} ∈ G_{0} such that H = a_{1} + K ∪ ... ∪ a_{k} + K. Let x ∈ H ∩ G_{1}. Then x = a_{i} + b where b ∈ K. Let f be as in Prop. 1.10, then 0 = f(x) = f(a_{i}) + f(b) = a_{i} + f(b). But f(b) ∈ K, so x ∈ K and H ∩ G_{1} = K ∩ G_{1}.

Now we show that T (𝔾_{1}) is a consequence of 𝕋_{1}. Let ϕ and ψ be a pair of pp-formulas from L_{0} which define subgroups H and K of G^{n} such that H ∩ G_{1}^{n} = K ∩ G_{1}^{n}. We have to show that the sentence ∀x̄ (ϕ(x̄) ↔ ψ(x̄)) is a consequence of 𝕋_{1}. Note first that if K ⊂ H and [H : K] is finite, this follows from the last axioms by an easy induction on n. But we can consider the pair K ∩ H ⊂ H and remark that [H : K ∩ H] is finite: indeed, if [H : K ∩ H] was infinite, then [H ∩ G_{1}^{n} : K ∩ H ∩ G_{1}^{n}] would be infinite. For the dimension sentences, suppose now that H and K are pp-definable subgroups of G such that K ∩ G_{1} ⊂ H ∩ G_{1}. Then the sentence for the index of [H ∩ G_{1} : K ∩ G_{1}] is a consequence of the sentence from 𝕋_{1} corresponding to the pair of pp-formulas ψ ∩ ϕ and ϕ.

Remark 1.14. Note that G_{1} contains no non trivial finite pp-definable subgroup. So, in 𝔾_{1}, the algebraic closure of the empty set is reduced to 0 and algebraic closure corresponds to definable closure.

Suppose now that 𝔾 is a model of 𝕋 and that 𝕊 is a model of the theory 𝕋_{1} := 𝕋_{1}(𝔾), and consider the L_{c}-structure 𝔾_{S} = 𝔾 ⊕ 𝕊 as in section 1.2, then:

Corollary 1.15. If 𝕊 is a model of 𝕋_{1} and 𝔾 is a model of 𝕋, then 𝔾_{S} is a model of 𝕋 and the map i from G into G_{S}, i(g) = (g, 0) is an L_{c}-elementary isomorphism (i.e. G ⊕ {0} is an L_{c}-elementary substructure of 𝔾_{S}).

Proof. Let ϕ and ψ be two pp-formulas such that ϕ(G) ⊂ ψ(G). Then ϕ(S) ⊂ ψ(S) (by equivalence sentences). If [ψ(G) : ϕ(G)] = k then ψ(S) = ϕ(S) (by 𝕋_{1}), so [ψ(G_{S}) : ϕ(G_{S})] = k. If [ψ(G) : ϕ(G)] = ∞ then [ψ(G_{S}) : ϕ(G_{S})] = ∞. Thus, 𝔾_{S} is a model of 𝕋, so G ⊕ {0} and 𝔾_{S} are elementarily equivalent, and it follows by pp-elimination in 𝕋 and Lemma 1.7 that G ⊕ {0} ≺ 𝔾_{S}.

2 The strongly minimal case

We remind the reader that a structure M is said to be strongly minimal if every definable subset of M (with parameters possibly in an elementary extension of M) is finite or co-finite. If M is strongly minimal, model theoretic algebraic closure (denoted acl) defines a pregeometry on M, in particular, for any X ⊂ M, the dimension of X (the cardinality of a maximal algebraically free subset in X) is well defined. We say that M is trivial, or has trivial pregeometry if, for all A ⊂ M, acl(A) = ⋃_{a∈A} acl{a}. We say that M is locally modular if for all algebraically closed X, Y ⊂ M, such that dim(acl(X ∩Y)) > 0, dim(X ∪Y) = dim(X)+dim(Y)−dim(X ∪Y). We will explicitly state the results we use about locally modular strongly minimal groups. For proofs and details we refer to [19], [4] or [15].

By a strongly minimal group, we mean, as usual, that G is a group with possibly extra structure, in a language L, which is strongly minimal as an L-structure.

Let G = (G, L) be a strongly minimal group in a language L. Let G_{0} := G ∩ acl(∅). A quasi-endomorphism of G is a connected subgroup H of G^{2}, definable over acl^{eq}(∅), different from G^{2}, such that the first projection of H is equal to G. It follows that H is strongly minimal. We define the kernel of H and the cokernel of H to be respectively:

KerH := {a ∈ G : (a, 0) ∈ H}, Coker(H) := {a ∈ G : (0, a) ∈ H}.

The cokernel of H is always finite, and if H is not trivial, that is if H ≠ G × 0, the kernel of H is finite. We denote by QS(G) the set of quasi-endomorphisms of G.

Remark: By strong minimality, if G_{0} is infinite, G_{0} is an elementary substructure of G and in that case, all quasi-endomorphisms of G are actually definable over G_{0}. In any case, if M_{0} is a prime model for T h(G), all quasi-endomorphisms are definable over M_{0}.

A quasi-endomorphism induces an endomorphism of G/G_{0}: if H ∈ QS(G), then {(a + G_{0}, b + G_{0}) : (a, b) ∈ H} is the graph of an endomorphism f_{H} of G/G_{0}. Furthermore the map which to every H ∈ QS(G) assigns the endomorphism f_{H} is a bijection from QS(G) onto the ring of the “quasi-definable” endomorphisms of G/G_{0}. The ring of endomorphisms of G/G_{0} induces the structure of a division ring on QS(G).

In the case of a locally modular group, the pregeometry on G defined by the relation of algebraic closure corresponds to the geometry of QS(G)-vector spaces. More precisely:

Fact 2.1. Let G be a locally modular strongly minimal group, let b, a_{1}, …, a_{n} ∈ G. then b ∈ acl(a_{1}, …, a_{n}) if and only if there are quasi-endomorphisms S_{1}, …, S_{n} and elements h_{1}, …, h_{n} ∈ G such that for each i, 1 ≤ i ≤ n,

(a_{i}, h_{i}) ∈ S_{i} and b − (h_{1} + … + h_{n}) ∈ G_{0}.

Any strongly minimal abelian structure 𝔸 is locally modular and by 1.6, all quasiendomorphisms of 𝔸 are definable over ∅.

By general results about one-based groups (see section 3.4), any locally modular strongly minimal group is “almost interdefinable” with a strongly minimal abelian structure. This will enable us at the end to reduce to the case of finitely axiomatizable abelian structures.

From now on in this section, 𝔾 is a strongly minimal abelian structure with constants, in the language L_{c}.

Consider, T (𝔾), the axiomatization of the theory of 𝔾 given in the previous sections, and 𝕋_{1} = 𝕋_{1}(𝔾), the associated theory.

Note that one can see directly from the axiomatizations of the form T (𝔾) when an abelian structure is strongly minimal: by pp-elimination 𝔾 will be strongly minimal if and only if 𝔾 is infinite and for any pp-definable subgroup H of G, H is finite or equal to G.

Lemma 2.2. The theory 𝕋_{1} is strongly minimal.

Proof. First a model of 𝕋_{1} must be infinite (the formula x = x is a pp-formula). Let ϕ be a pp-definable subgroup of G. By strong minimality of 𝔾, either ϕ is finite, or ϕ is equal to G. Hence in any model ℍ of 𝕋_{1}, ϕ is trivial or ϕ = H. By pp-elimination, 𝕋_{1} is also strongly minimal.

We also know (see Remark 1.14) that in any model of 𝕋_{1}, acl(∅) = {0} and acl = dcl. It follows easily that if 𝔾_{1} is a model of 𝕋_{1} and if K_{1} is the division ring of quasiendomorphisms of 𝔾_{1}, the structure on 𝔾_{1} is exactly the K_{1}-vector space structure. But we want to check that K_{1} = K, where K is the division ring of quasi-endomorphisms of 𝔾.

Let us recall the definition of interdefinability:

Definition 2.3. Let M_{1} = (M, L_{1}) and M_{2} = (M, L_{2}) be, respectively, L_{1} and L_{2}- structures with the same universe M. Let A ⊂ M, we say that M_{1} is A-definable in M_{2} if every A-definable subset in M_{1} is A-definable in M_{2}, equivalently, if every symbol in the language L_{1} is A-definable in M_{2}. We say that M_{1} and M_{2} are A-interdefinable if each M_{i} is A-definable in the other, equivalently, if M_{1} and M_{2} have the same A-definable subsets.

Proposition 2.4. Let K be the division ring of quasi-endomorphisms of 𝔾. Then, if 𝕊 is any model of 𝕋_{1}, 𝕊 carries a K-vector space structure with which it is ∅-interdefinable.

Proof. Let 𝔾′_{S} = 𝔾′ ⊕ 𝕊, where 𝔾′ is a countable elementary substructure of 𝔾 (take for G′, G_{0} = acl(∅), if it is infinite) and 𝕊 is any model of 𝕋_{1}. Then 𝔾′_{S} is a model of 𝕋, the theory of 𝔾 (1.15). Let ϕ_{r}(x, y) be a pp-formula which defines the quasiendomorphism r ∈ K. By the axioms in 𝕋_{1}, in S × S, ϕ_{r}(x, y) defines a subgroup such that its first projection is equal to S, which is strict and connected, hence which is a quasi-endomorphism of S. The kernel, {y ∈ S : S ⊧ ϕ_{r}(0, y)}, must be trivial, as well as the co-kernel. So ϕ_{r}(x, y) defines an endomorphism of S. It follows easily that the set of pp-formulas (ϕ_{r} : r ∈ K) induce a K-vector space structure on S.

Now, it suffices to prove that any two tuples, a_{1}, ..., a_{n} and b_{1}, ..., b_{n}, of S have the same L_{0}-type if they have the same K-vector space type.

Claim. Let a, b, a_{1}, .., a_{n} ∈ S be such that a is K-linearly independent from a_{1}, ..., a_{n} and b is also K-linearly independent from a_{1}, ..., a_{n}, then there exist an L_{0}-automorphism of S which sends a to b and fixes a_{1}, ..., a_{n}.

Proof. In the structure 𝔾′_{S}, (0, a) ∉ acl((G′ ⊕{0})∪{(0, a_{1}), ..., (0, a_{n})}) (in the language L_{c}): otherwise, there would be r_{1}, ..., r_{n} ∈ K, (x_{1}, y_{1}), ..., (x_{n}, y_{n}) ∈ G′_{S} and g ∈ G′ such that

(0, a) = (x_{1}, y_{1}) + ... + (x_{n}, y_{n}) + (g, 0) and G′_{S} ⊧ ϕ_{r_{i}}((0, a_{i}), (x_{i}, y_{i})).

But then, a = y_{1} + ... + y_{n} = r_{1}a_{1} + .... + r_{n}a_{n}. Similarly (0, b) ∉ acl((G′ ⊕ {0}) ∪ {(0, a_{1}), ..., (0, a_{n})}). Since 𝔾′_{S} is strongly minimal and G′ ⊕ {0} ≺ 𝔾′_{S}, there exists an automorphism τ of 𝔾′_{S} which sends (0, a) to (0, b) and fixes pointwise G′ ⊕ {0} and (0, a_{1}), ..., (0, a_{n}). From τ we construct easily an L_{0}-automorphism of S which sends a to b and fixes a_{1}, ..., a_{n}: just let σ(s) := s′ if τ(0, s) = (x, s′).

Let a_{1}, ..., a_{n} and b_{1}, ..., b_{n} be two tuples of S which have the same K-vector space type. Assume that a_{1}, .., a_{k} are K-linearly independent and for every j > k, a_{j} is dependent on a_{1}, .., a_{k}. By the previous claim, there is an L_{0}-automorphism σ of S which sends a_{1}a_{2}...a_{k} on b_{1}....b_{k}. But, then for every j > k, σ(a_{j}) = b_{j}.

Remark. If 𝔾 is such that G_{0} = acl(∅) = 0, then 𝕋 = 𝕋_{1} and 𝔾 itself has the structure of a K-vector space. In that case, it will follows directly that 𝕋 is finitely axiomatizable if and only if the theory of infinite K-vector spaces is finitely axiomatizable, and hence if and only if (see section 3.2.2) K is finitely presented as a ring.

3 Application to finite axiomatizability

In [17] Paljutin shows that “There exists a finitely axiomatizable, not locally finite categorical quasi-variety if and only if one of the following conditions hold:

1) there exists an infinite finitely presented ring which is a division ring;

2) there exists an infinite finitely presented group with a finite number of elements g_{1}, …, g_{n} such that every non trivial cyclic subgroup of G intersects one of the conjugacy classes of the elements g_{1}, …, g_{n}.”

The proof proceeds by showing first that, if there is a such a finitely axiomatizable quasi-variety (a quasi-variety is the class of models of a set of universal Horn sentences), then there exists one which is “standard”, where the standard quasi-varieties are either K-vector spaces for a division ring K, or the Cayley graph of a group. Then he shows that if the quasi-variety of K-vector spaces is finitely axiomatizable and not ω-categorical, K must satisfy 1), and that if the Cayley graph of G is finitely axiomatizable and not ω-categorical, then G must satisfy 2) (a proof of this part, due to M.A. Taitslin and Yu.E. Shimarev had already appeared in [1]).

The existence of such a ring and such a group are both still open. The existence of an infinite finitely presented group with finitely many conjugation classes is a well-known long standing open question, but the existence of the a priori weaker condition required in 2) is also open.

Concerning the existence of a ring satisfying 1), it seems that it is already unknown whether there exists an infinite finitely generated ring which is a division ring. One can only easily see that such a division ring cannot be commutative (see section 3.2.2 below).

For some years now A. Ivanov has been working on the conjecture that the existence of any finitely axiomatizable strongly minimal trivial set must imply the existence of a group satisfying 2) and has proved some partial results ([12, 13]). The idea behind this conjecture is that such a group should appear as a subgroup or a quotient of the automorphism group of a connected component of the trivial strongly minimal set.

As explained in the introduction, Hrushovski then showed in [10] that any finitely axiomatizable ℵ_{1}-categorical theory must be locally modular and suggested the conjecture that if the theory has non trivial pregeometry, then the associated division ring of quasi-endomorphisms must satisfy 1).

In order to be quite self-contained on the subject of finite axiomatizability, and because sometimes a certain confusion arises on what exactly is meant by finite axiomatizability (in the case of an infinite language, for example), we will present in the next section precise definitions and basic transfer properties. For the same reasons, in section 3.2, we will present a detailed exposition of the two “standard” cases.

3.1 Transferring finite axiomatizability

We are going to need to transfer the property of being finitely axiomatizable through various changes of languages and interpretations, and to be quite precise when we do it. We have unfortunately not found a completely adequate reference for our purpose, which we could have simply quoted or referred to. Most of what follows appears in various places under slightly different forms. The closest references for bi-interpretability can be found in [2] or, more recently, in [19]. In both cases the notions were used in the context of quasi-finitely axiomatizable ω-categorical theories.

First, we need to recall precisely what it means for a theory in an infinite language to be finitely axiomatizable.

¿From now on, when we use the word theory, we mean a consistant set of axioms which is closed under deduction.

Definition 3.1. Let L be an infinite language and T a theory in L. We say that T is finitely axiomatizable if there is a finite sub-language L_{0} of L such that any model M of T is interdefinable (Definition 2.3) with its reduct to L_{0} and the theory T_{L_{0}}, the restriction of T to the language L_{0}, is finitely axiomatizable in L_{0}. We then say that T is finitely axiomatizable in the finite language L_{0}.

The following easy lemma will be useful later:

Lemma 3.2. Let T be a theory in a language L, let L_{0} be a sub-language of L and let Σ be a subset of sentences from T with the following properties:

– for every predicate symbol R(x̄) in the language L, Σ includes a sentence of the form ∀x̄ (ϕ_{R}(x̄) ↔ R(x̄)) where ϕ_{R}(x) is a formula in L_{0},

– for every function symbol f(x̄) in L, Σ includes a sentence of the form ∀x̄∀y (ϕ_{f}(x̄, y) ↔ f(x̄) = y) where ϕ_{f}(x̄, y) is a formula in L_{0},

– for every constant symbol c in L, Σ includes a sentence of the form ∀x (ϕ_{c}(x) ↔ x = c) where ϕ_{c}(x) is a formula in L_{0}.

Then T is axiomatized by Σ ∪ T_{L_{0}}.

Finite axiomatizability is easily seen to transfer through interdefinability.

Lemma 3.3. Let M_{1} = (M, L_{1}) and M_{2} = (M, L_{2}) be two structures on the same domain M which are interdefinable. Then T h(M_{1}) is finitely axiomatizable if and only if T h(M_{2}) is finitely axiomatizable.

First some basic remarks about extending the language while keeping finite axiomatizability:

Lemma 3.4. (i) Let M = (M, L) be a structure in a finite language L. If E is a ∅-definable equivalence relation in M then the theory of M has a finite axiomatization if and only if the theory of the reduct of M^{eq}, M_{E} = (M, M^{n}/E, L, f_{E}) has a finite axiomatization.

(ii) Let M = (M, L) be a structure in a finite language L. Let ā be a tuple of M such that the type of ā is isolated. Then the theory of M has a finite axiomatization if and only if the theory of Mā = (M, L, ā) has a finite axiomatization.

We now check that one can reduce questions of finite axiomatizability for ℵ_{1}-categorical one-based groups to the case of abelian structures.

Recall the fundamental properties of one-based groups (the reader can take Property 1 as a definition for one-based groups)

Fact 3.5. [11] 1. A group G is one-based if and only if, for every n ≥ 1, every definable subset of G^{n} is a Boolean combination of definable cosets of connected definable subgroups of G^{n}.

2. A one-based group is definably abelian by finite, that is, has a definable normal abelian subgroup of finite index.

3. Let H be a one-based group. Let S ⊂ H^{n} be a definable connected subgroup. Then S is definable over acl^{eq}(∅).

Corollary 3.6. Let G be an ℵ_{1}-saturated one-based group. Fix M_{0} a countable elementary sub-model of G. Let A ⊂ G; any A-definable subset of G^{n} is a Boolean combination of A-definable cosets of some M_{0}-definable connected subgroups.

Now let G = (G, L) be an ℵ_{1}-saturated connected one-based group. Fix some countable elementary sub-model of G, M_{0}. Let (H_{i})_{i∈I} be the family of all connected M_{0}- definable subgroups in ⋃_{n≥1} G^{n}. Let 𝔾 be the following abelian structure:

𝔾 =< G, +, −, 0, (H_{i})_{i∈I}, (m)_{m∈M_{0}} > .

Then it follows from Fact 3.5 and Corollary 3.6 that G and 𝔾 are M_{0}-interdefinable.

Corollary 3.7. Let G be an ω-stable one-based connected group such that T h(G) is finitely axiomatizable in a finite language L. Then there is a finitely axiomatizable abelian structure with constants 𝔾, which is interdefinable with a finitely axiomatizable expansion of G by finitely many constants.

Proof. Without loss of generality, suppose that G is ℵ_{1}-saturated. Consider M_{0} ⪯ G, the prime model of T h(G), by ω-stability. Consider the abelian structure 𝔾 described above which is M_{0}-interdefinable with G. As the language L is finite, we can choose a finite family H_{1}, …, H_{k} and a finite sequence m_{0}, …, m_{n} of elements from the prime model M_{0}, such that every symbol from the language L can be defined in the restriction 𝔾_{1} of 𝔾 to the finite language

L_{1} := {+, −, 0, (H_{i})_{1≤i≤k}, {m_{0}, …, m_{n}}}

and such that every H_{i} is definable in G over {m_{0}, …, m_{n}}. Now add {m_{0}, …, m_{n}} as new constants to the language L of G. As M_{0} is atomic, the tuple (m_{0}, …, m_{n}) has isolated type over ∅ in L. By Lemma 3.4, the theory of the expansion G′ of G to L′ := L ∪ {m_{0}, …, m_{n}} remains finitely axiomatizable. Now the structures G′ and 𝔾_{1} are interdefinable (over ∅). Hence by Lemma 3.3 T h(𝔾_{1}) is finitely axiomatizable.

We will also need to check that an ℵ_{1}-categorical group is finitely axiomatizable if and only if its connected component is. This is a particular case of the transfer of finite axiomatizability by bi-interpretability.

Recall the definition of bi-interpretability from [2], or [19]:

Definition 3.8. Let M = (M, L_{1}) and N = (N, L_{2}). We say that M is ∅-interpretable in N if there is a ∅-definable subset U of N^{n}, and a surjective map f, from U onto M such that:

– the equivalence relation E_{f} on U × U defined, for (a, b) ∈ U × U, by f(a) = f(b) is ∅-definable in N,

– for every k-ary relation symbol R in L_{1}, the subset R_{f} of U^{k}, R_{f} := {(a_{1}, …, a_{k}) ∈ U^{k}; M ⊧ R(f(a_{1}), …, f(a_{k}))} is ∅-definable in N,

– for every k-ary function symbol h in L_{1}, the subset Gr_{f}(h) := {(a_{1}, …, a_{k}, b) ∈ U^{k+1} ; M ⊧ g(f(a_{1}), …, f(a_{k})) = f(b)} is ∅-definable in N,

– for every constant symbol c in L_{1}, the subset c_{f} := {a ∈ U : M ⊧ f(a) = c} is ∅-definable in N.

It follows that f induces an isomorphism of L_{1}-structures between U/E_{f} (subset of N^{eq}) and M.

If M is interpretable in N, via the surjective map f from U ⊂ N^{k} onto M, and N is interpretable in Q = (Q, L_{3}), via the surjective map g from V ⊂ Q^{n} onto N, let W := {(q_{1}, …, q_{k}) ∈ (Q^{n})^{k} : (g(q_{1}), …, g(q_{k})) ∈ U}. We denote by f ∘ g the obvious induced map from W ⊂ S^{nk} onto M. One checks easily that f ∘ g is an interpretation of M in Q.

If M is interpretable in N, via f, and N is interpretable in M via g, we say that M and N are bi-interpretable if f ∘ g is a ∅-definable map in M and g ∘ f a ∅-definable map in N.

We leave the checking of the following lemma to the reader.

Lemma 3.9. Let M be an L_{1}-structure and N an L_{2}-structure which are bi-interpretable. Then T h(M) is finitely axiomatizable if and only if T h(N) is.

We recall the definition of an induced structure on a definable subset:

Definition 3.10. Let M be an L-structure and D be a ∅-definable subset of M^{n}. The induced structure from M on D is the structure < D, (P_{ϕ})_{{ϕ formula in L}} >, where, for ϕ(x_{1}, …, x_{k}), |x_{j}| = n, P_{ϕ} is a predicate of arity k which is interpreted on D by the set D^{k} ∩ ϕ(M^{nk}).

Lemma 3.11. Let M be an L-structure which is the union of a finite definable partition, that is, M = M_{1}∪…∪M_{n}, where, for 1 ≤ i < j ≤ n, M_{i} is ∅-definable and M_{i}∩M_{j} = ∅. Suppose furthermore that for each i > 1 there is a ∅-definable bijection, f_{i} from M_{1} onto M_{i} and that L contains n constant symbols {c_{1}, …, c_{n}} which are interpreted in M by distinct elements of M_{1}. Consider M_{1} together with its induced structure from M, denoted M_{1}. Then M and M_{1} are bi-interpretable.

Proof. Let U := M_{1} × {c_{1}, …, c_{n}} ⊂ M_{1}^{2} and let f : U ↦ M, f(x, c_{i}) = f_{i}(x), where f_{1} is the identity on M_{1}. Then f gives an (injective) interpretation of M into M_{1}. Indeed, for any k-ary predicate symbol R from L, the set R_{f} = {(a_{1}, …, a_{k}) ∈ U : M ⊧ R(f(a_{1}), …, f(a_{k})} is a basic predicate in the language of the induced structure, hence certainly definable. Similarly for the other symbols of L.

Let g : M_{1} ⊂ M ↦ M_{1} be the identity. Let W := {(x, y) ∈ U : f(x, y) ∈ M_{1}} = M_{1} × {c_{1}}. Then g ∘ f : W ↦ M_{1} is an interpretation of M_{1} into itself. For (x, c_{1}) ∈ U, g ∘ f(x, c_{1}) = x is the first projection and is hence ∅-definable in M_{1}.

Let W′ := {(x, y) ∈ M_{1}×M_{1} : (g(x), g(y)) ∈ U}. Then W′ = U = M_{1}×{c_{1}, …, c_{n}}, and f ∘ g(x, c_{i}) = f_{i}(x) is ∅-definable in M.

Corollary 3.12. Let G = (G, L) be an ω-stable group in a language L. Let G^{0} be the connected component of G, G^{0}, with the induced structure from G. Then T h(G) is finitely axiomatizable if and only if T h(G^{0}) is finitely axiomatizable.

Proof. By ω-stability, G^{0} is ∅-definable in G and has finite index in G. By ω-stability again, there is a prime model G_{1}, G_{1} ⪯ G, atomic over ∅. Choose a_{1}, …, a_{n} in this prime model such that G = a_{1}G^{0} ∪ … ∪ a_{n}G^{0}, with a_{1} = 1. Choose c_{1}, … c_{n} distinct elements from G_{1}^{0}, the connected component of G_{1}. Then the type of the tuple a_{1}, …, a_{n}, c_{1}, …, c_{n} is isolated over the empty set. Let T denote the complete theory of G in the language L together with {a_{1}, …, a_{n}, c_{1}, …, c_{n}} added as new constants. By 3.4, T h(G) (in L) is finitely axiomatizable if and only if T is. Similarly, let T_{0} be T h(G^{0}) in the language with {c_{1}, …, c_{n}} added as new constants, then T h(G^{0}) is finitely axiomatizable iff T_{0} is. We are now in the situation of Lemma 3.11: G is the union of a finite ∅-definable partition a_{1}G^{0} ∪ … ∪ a_{n}G^{0}, and for each i, there is a ∅-definable bijection f_{i} from a_{i}G^{0} onto a_{1}G^{0} = G^{0}, f_{i}(a_{i}g) = g. The result then follows by 3.11 and 3.9.

3.2 The classical examples

Before we start on the description of the two emblematic examples, we would like to draw the reader’s attention to the following: if T is a theory in L which is finitely axiomatizable in a finite sub-language L_{0} of L then it is certainly finitely axiomatizable in every finite sub-language L_{1} of L containing L_{0} (Lemma 3.2), but one should be a little careful. For example suppose that T is a complete theory in an infinite language L, which is finitely axiomatizable in a finite sub-language L_{0} of L. Let 𝔽_{0} be a finite set of axioms for T_{L_{0}}, and let Σ be an arbitrary infinite set of axioms in L for the complete theory T. By compactness some finite subset Σ_{1} of Σ will axiomatize T_{L_{0}}. But if L_{1} is the finite sub-language of L containing all symbols appearing in Σ_{1}, there is no reason that Σ_{1} ⊢ T_{L_{1}}, or equivalently there is no reason for Σ_{1} to axiomatize a complete theory in the language L_{1}. This explains the care taken in identifying the right set of axioms in the following proofs.

3.2.1 The trivial example

First, recall that for any non trivial group G, the theory T_{G} which describes G acting semi-regularly (the stabilizer of every element is trivial) on an infinite set, in the language L_{G} := {g : g ∈ G}, where each g is a unary function symbol, is strongly minimal, eliminates quantifiers and has trivial geometry. The theory T_{G} is ω-categorical if and only if G is finite. If G is infinite, the Cayley graph of G (that is the regular action of G on itself by left multiplication) is a model of T_{G}. The theory T_{G} can be axiomatized by the following set of axioms, Σ_{G}, if G is infinite:

– ∀x 1(x) = x; – ∀x g(x) ≠ x, for every g ≠ 1 ∈ G;

– ∀x g(h(x)) = r(x), for every g, h, r ∈ G such that gh = r. If G is finite, then T_{G} can be axiomatized by Σ_{G} together with the scheme for infinity. Note that, for any model M of T_{G}, for any a ∈ M, the definable closure of {a} in M in the language L_{G} is the G-orbit of a. Now suppose that we have a presentation of G, i.e. that G is isomorphic to the free group on S = {s_{i}; i ∈ I} modded out by a normal subgroup P. Then the theory T_{G} is clearly interdefinable with the following theory in the language L_{S} = {s : s ∈ S}, which we denote by Σ_{S}: let W be the set of words on S,

– ∀x w(x) = x for every w ∈ P,

– ∀x w(x) ≠ x for every w ∈ W \ P.

Suppose that G is an infinite finitely presented group G, with a finite number of conjugation classes C_{1}, … C_{k} such that every non trivial cyclic subgroup of G intersects one of the C_{i}’s. Then T_{G} is finitely axiomatizable (in the sense of Definition 3.1). Indeed, choose F = {g_{1}, …, g_{n}} ⊂ G such that: for every j, g_{j} ≠ 1, F generates G, there is a finite set P_{0} of words on F which generates the presentation of G, P, F is closed under inverse, and for every g ∈ G \ {1}, there is some m > 0 such that g^{m} is conjugate to one of the g_{j}’s. Let Σ_{F} be the (complete) set of axioms described above, in the finite language L_{F} := {g_{1}, …, g_{n}}, which is interdefinable with T_{G}. Consider Σ_{0}, the following finite subset of Σ_{F}:

– ∀x g_{j}(x) ≠ x, for every j, 1 ≤ j ≤ n,

– ∀x w(x) = x, for every w ∈ P_{0}.

We must check that Σ_{0} is an axiomatization for Σ_{F}. If w ∈ P, then, for all x, w(x) = 1 as P is the normal subgroup generated by P_{0}. If g is any word on F, and g ∉ P, we must check that for all x, g(x) ≠ x. By assumption, there are m > 0, g_{j} ∈ F and h ∈ G, such that g^{m} = h^{−1}g_{j}h. If g(x) = x for some x, then g^{m}(x) = x = h^{−1}(g_{j}(h(x)), hence h(x) = g_{j}(h(x)). But this contradicts Σ_{0}.

Conversely, suppose that G is infinite and that the theory T_{G} is finitely axiomatizable. Let F ⊂ G, be finite such that T := T_{G} is finitely axiomatizable in the sub-language L_{F} := {f; f ∈ F }, that is, such that any model of T_{G} is interdefinable with its reduct to L_{F}, and the (complete) theory T_{G}|L_{F} is finitely axiomatizable. Let H be the subgroup of G generated by F. Then T_{G}|L_{H} contains the theory T_{H}, which is complete, hence it is equal to T_{H}. In the language L_{G}, G which is a model of T_{G}, is equal to the definable closure of the identity element 1. Similarly, H is, in L_{H} the definable closure of 1. By interdefinablity of L_{G} with L_{F}, G is also equal to the definable closure of 1 in L_{F}, hence also in L_{H}. It follows that G = H. So we know that G is finitely generated, hence isomorphic to the free group on a finite set of generators S, which we suppose closed under inverse, modded out by a normal subgroup P. Let W be the set of all words on F. Pass to the theory T_{S} (axiomatized by Σ_{S}) in the finite language {s : s ∈ S}, which is interdefinable with T_{G}, By finite axiomatizability, there is a finite subset W_{0} of W (the set of words on S) such that Σ_{S} can be axiomatized by Σ_{W_{0}}:

– ∀x w(x) = x for every w ∈ W_{0} ∩ P,

– ∀x w(x) ≠ x for every w ∈ W_{0} \ P. We can suppose that, for every s ∈ S, ss^{−1} ∈ W_{0}. Let N be the normal subgroup generated by W_{0} ∩ P in D, the free group on S. By construction N ⊂ P. The Cayley graph of D/N, in the language L_{S} is a model of Σ_{W_{0}}, hence is interdefinable with a model of Σ_{S}. It follows that N = P and G ≅ D/N. Let h ∈ G, h ≠ 1, let H denote the subgroup generated by h in G, and G/H the set of left cosets, equipped with an L_{G} structure by the left action of G. As the action of G is not semi-regular on G/H, G/H is not a model of Σ_{S}, hence by finite axiomatizability, it is not a model of Σ_{W_{0}}. So there is some g ∈ W_{0} \ N and some coset aH such that g(aH) = aH, that is, such that a^{−1}ga = h^{n} for some integer n. So any non trivial element h has a power which is conjugate to one of the g′s in W_{0}.

3.2.2 Vector spaces

Let K be any countable division ring. Let L_{K} be the usual language for K-vector spaces, L_{K} := {+, −, 0, (k)_{k∈K}}, where k is a unary function interpreted as scalar multiplication by the element k. Consider T_{K} the theory of all infinite K-vector spaces in L_{K}. The theory T_{K} is ℵ_{1}-categorical and is totally categorical if and only if K is finite.

Suppose that K is an infinite division ring which is finitely presented as a ring. Then the complete theory of K-vector spaces is finitely axiomatizable in the following way: let F be a finite subset of K, which generates K as a ring and such that there is a finite set of terms in F, P, which generates the presentation of K (a two-sided ideal J, such that K is isomorphic to the quotient fo the free ring generated by F BY J). Then T_{K} is finitely axiomatized in L_{F} := {+, −, 0, 1, (f)_{f∈F} } by

– axioms for abelian groups

– ∀x 1(x) = x,

– ∀x ∀y f(x + y) = f(x) + f(y), for every f ∈ F

– ∀x w(x) = 0, for every w ∈ P.

For the converse, we now suppose that the theory of infinite K-vector spaces is finitely axiomatizable. By the classical results on the non finite axiomatizability of totally categorical theories, the assumption forces K to be infinite.

Proposition 3.13. Let K be an infinite division ring. If the theory of K-vector spaces is finitely axiomatizable then K is finitely presented as a ring.

Proof. Let T_{K} be the theory of non trivial K-vector spaces, in the usual language L_{K} = {0, +, −, k : k ∈ K}.

Let X be a finite subset of K such that T_{K} is finitely axiomatizable in the finite language L_{X} := {0, +, −, k : k ∈ X}.

Claim. K is generated as a skew field by X.

Proof. Let K_{0} be the subfield of K generated by X. Then the theory T_{K_{0}} of infinite K_{0}- vector spaces is a subset of T_{|L_{K_{0}}} and since T_{K_{0}} is complete, they are equal. Now, consider K as a K-vector space. Then K = dcl_{L_{K}}(1_{K}) = dcl_{L_{X}}(1_{K}) = dcl_{L_{K_{0}}}(1_{K}) = K_{0}.

Denote by S_{K} the classical axiomatization of T_{K} :

1. ∃x x ≠ 0;

2. axioms for abelian groups;

3. ∀x 1_{K}(x) = x;

4. ∀x∀y k(x + y) = k(x) + k(y), (k ∈ K);

5. ∀x k(x) + k′(x) = k′′(x), (k, k′, k′′ ∈ K, k′′ = k + k′);

6. ∀x k(k′(x)) = k′′(x), (k, k′, k′′ ∈ K, k′′ = kk′).

If A is a subset of K we will denote by S_{A} the subset of sentences of S_{K} in the language L_{A} := {0, +, −, k : k ∈ A}. A priori, S_{X} does not give an axiomatization of the complete theory T_{L_{X}}, but since there is some finite axiomatization of T_{L_{X}} by assumption, there exists by compactness a finite subset Y containing X such that S_{Y} implies T_{L_{X}}.

We are going to enlarge Y in order that S_{Y} implies the complete theory T_{L_{Y}}. First we define the depth of elements of K relatively to X. We assume that 1_{K}, O_{K} ∈ X and we define by induction a sequence (W_{i})_{i∈ω} of subsets of K such that W_{0} := X and

W_{i+1} := {k ∈ K : k = −k_{1} or k = k_{1}^{−1} or k = k_{1} + k_{2} or k = k_{1}k_{2} for k_{1}, k_{2} ∈ W_{i}}.

Then K = ∪_{i∈ω}W_{i} since X generates K as a skew field K. We define the depth of k ∈ K as the smallest integer n such that k ∈ W_{n}. Now by an easy induction, we can enlarge Y so that it remains finite and for each k ∈ Y, if the depth of k is n + 1 then there exists k_{1}, k_{2} ∈ Y of depths at most n such that k = −k_{1}, or k = k_{1}^{−1}, or k = k_{1} + k_{2} or k = k_{1}k_{2}.

Claim. Then T_{L_{Y}} is axiomatized by S_{Y}.

Proof. We choose, by induction on the depth of elements of Y, for each k ∈ Y, a formula ϕ_{k}(x, y) ∈ L_{X} such that S_{Y} ⊢ ∀x∀y (kx = y) ↔ ϕ_{k}(x, y) (we know by assumption that there is such a formula for which T ⊢ ∀x∀y (kx = y) ↔ ϕ_{k}(x, y), but we want one such that the equivalence can be deduced from S_{Y}). If k ∈ Y has depth 0 (i.e. k ∈ X), then we let ϕ_{k}(x, y) := (kx = y). Assume that we have chosen a formula ϕ_{k} for each k ∈ Y of depth less or equal to n. Let k ∈ Y have depth n + 1. Then there exist k_{1}, k_{2} ∈ Y, of depth at most n, such that at least one of the following cases occur:

• k = −k_{1}; in this case we let ϕ_{k}(x, y) := ϕ_{k_{1}}(x, −y),

• or k = k_{1}^{−1}; in this case we let ϕ_{k}(x, y) := ϕ_{k_{1}}(y, x), • or k = k_{1} + k_{2}; in this case we let

ϕ_{k}(x, y) := ∃t_{1}∃t_{2} (ϕ_{k_{1}}(x, t_{1}) ∧ ϕ_{k_{2}}(x, t_{2}) ∧ (y = t_{1} + t_{2})) ,

• or k = k_{1}k_{2}; in this case we let

ϕ_{k}(x, y) := ∃t (ϕ_{k_{2}}(x, t) ∧ ϕ_{k_{1}}(t, y)) .

Since T_{K} is finitely axiomatizable in the language L_{X} and for each k ∈ Y, T_{L_{Y}} ⊢ ∀x∀y (kx = y) ↔ ϕ_{k}(x, y), the complete theory T_{L_{Y}} is axiomatized by

T_{L_{X}} ∪ {∀x∀y (kx = y) ↔ ϕ_{k}(x, y) : k ∈ Y }.

It follows that S_{Y} is an axiomatization of T_{L_{Y}}.

Now, we are going to prove that K is isomorphic to the finitely presented ring A, given by the set of generators {k : k ∈ Y } and the presentation:

{1_{K}−1}∪{k_{1}+k_{2}−k_{3} : k_{1}, k_{2}, k_{3} ∈ Y; k_{3} = k_{1}+k_{2}}∪{k_{1}k_{2}−k_{3} : k_{1}, k_{2}, k_{3} ∈ Y; k_{3} = k_{1}k_{2}}.

Remark that every non trivial A-module is a model of S_{Y} as an L_{Y} -structure, hence any two non trivial A-modules are elementarily equivalent in the language L_{Y}. Furthermore, any non trivial A-module has a canonical expansion to a K-vector space : by assumption, for each k ∈ K, there is a formula θ_{k} ∈ L_{X} ⊂ L_{Y}, such that T ⊢ ∀x∀y (kx = y) ↔ θ_{k}(x, y). Define , for k ∈ K, m, n ∈ M, km = n iff M ⊧ θ_{k}(m, n).

Let ψ be the canonical morphism from A to K which sends each generator k ∈ Y to k ∈ K. The morphism ψ is injective: consider the A-module structure on K given via ψ, i.e. define ax := ψ(a)x. As A-modules, K and A are elementarily equivalent. In A, if a ≠ 0, then for some x, ax ≠ 0, hence this is also true in K, which implies that ψ(a) ≠ 0.

Hence A has no zero divisors. Again by completeness of the theory of non trivial A-modules, this implies that in all non trivial A-modules, if a ≠ 0 ∈ A, if x ≠ 0, then ax ≠ 0. This implies that A is a division ring: if a ∈ A\{0} was not left invertible, A/Aa would be a non-trivial A-module satisfying that ax = 0 for some x ≠ 0 (x = 1 + Aa). Since X ⊂ ψ(A), X generates K as a skew field and ψ(A) is a skew field, we obtain that ψ(A) = K.

Remark. As we have mentioned above, it seems to be an open question whether there exists an infinite division ring which is finitely generated as a ring. It is easily seen, though, that there is no such commutative division ring: Let K be a field which is finitely generated as a ring, and let k denote its prime field (k = 𝔽_{p} or k = ℚ). As K is finitely generated as a ring over k, then K is contained in k^{alg}, the algebraic closure of k. If K has characteristic p > 0, then K = 𝔽_{p}[a_{1}, …, a_{n}] is finite. Otherwise k = ℚ, and there are a_{1}, …, a_{n} ∈ ℚ^{alg}, such that K = ℤ[a_{1}, …, a_{n}]. In that case, for some integer m > 0, the a_{i}’s are entire over A := ℤ[1/m] and K is finitely generated as an A-module. As A is Noetherian, K is Noetherian as an A-module, and ℚ, as an A-submodule, must also be finitely generated, which is impossible.

3.3 Finitely axiomatizable strongly minimal abelian structures

We suppose that 𝔾 is a strongly minimal abelian structure in a finite language L_{c} = L_{0} ∪ {c ∈ C} such that its theory 𝕋 is finitely axiomatizable. Recall from corollary 1.4 that 𝕋 is axiomatized by the set of sentences T (𝔾) together with the pp-type of the constants. Let 𝔹 be a finite axiomatization of 𝕋 which consists of a finite subset 𝔸 of T (𝔾) together with a finite subset of the pp-type of the constants. Denote by 𝔸_{1} the following finite subset of 𝕋_{1} (as defined in section 1.8):

- the axioms for abelian groups,

- for each original predicate H from L_{0}, H is a subgroup,

- the equivalence sentences in 𝔸,

- [ϕ : ψ] ≥ k for every k and for every pair of pp-formulas such that in 𝔾, ψ(G) ⊂ ϕ(G) ⊂ G, [ϕ(G) : ψ(G)] is infinite and the sentence [ϕ : ψ] ≥ k is in 𝔸,

- [ϕ : ψ] = 1 for every pair of pp-formulas such that in 𝔾, ψ(G) ⊂ ϕ(G) ⊂ G and for some integer k, the sentence [ϕ(G) : ψ(G)] = k is in 𝔸,

- ∃x x ≠ 0.

Lemma 3.14. For every model 𝔾 of 𝕋 and every model 𝕊 of 𝔸_{1}, the L_{c}-structure 𝔾_{S} (= 𝔾 ⊕ 𝕊 as in section 1.2) is a model of 𝔹 and so, of 𝕋. Moreover, G ⊕ {0} is an elementary submodel of 𝔾_{S}.

Proof. The proof that 𝔾_{S} is a model of 𝔹 is exactly similar to the proof of 1.15. One needs to check the dimensions only for the pp-subgroups ϕ, ψ such that a dimension sentence of the form [ϕ : ψ] ≥ n (if it is infinite) or [ϕ : ψ] = n (if it is finite) appears in 𝔸.

We are going to show that 𝔸_{1} gives an axiomatization for the complete theory 𝕋_{1}. It suffices to show that every model of 𝔸_{1} is infinite:

Lemma 3.15. If all models of 𝔸_{1} are infinite then 𝔸_{1} is a finite axiomatization of 𝕋_{1}.

Proof. We show that any two models of 𝔸_{1} of cardinality ℵ_{1} are isomorphic, then if 𝔸_{1} has no finite models, it is complete and hence axiomatizes 𝕋_{1}. Let 𝔾 be a countable model of 𝕋. (One can choose acl(∅) if it is infinite.) Let 𝕊_{1} and 𝕊_{2} be two models of 𝔸_{1} of cardinality ℵ_{1}. Then by strong minimality, as G ⊕ {0} is algebraically closed in 𝔾_{S_{i}}, there is an isomorphism between 𝔾_{S_{1}} and 𝔾_{S_{2}} which is the identity on G ⊕ {0}. From this isomorphism one induces easily an isomorphism between 𝕊_{1} and 𝕊_{2}.

Proposition 3.16. The ring of quasi-endomorphisms of 𝔾 is infinite.

Proof. Suppose not. Let K denote the ring of quasi-endomorphisms, then K = 𝔽_{q} and G has bounded exponent. In particular every finitely generated subgroup of G is finite. We are going to construct a finitely generated subgroup of G which is a model of 𝕋, contradicting the completeness of 𝕋.

First, we add the quasi-endomorphisms as predicates to the language: for each α ∈ 𝔽_{q}, denote by H_{α} the corresponding quasi-endomorphism, which is a strongly minimal subgroup of G^{2}, definable over ∅ by Lemma 1.6, such that its first projection is equal to G. We add to L_{c} a predicate Ĥ_{α} for each α ∈ 𝔽_{q}. This preserves the finite axiomatizability of 𝔾. So we can assume that the language L_{c} already contains the quasi-endomorphisms as predicates. Now, we also, if necessary, add finitely many new predicates for some pp-definable subgroups which appear in the axiomatization 𝔹, so that 𝕋 has a finite axiomatization containing only sentences of the following type where X, Y and Z are amongst the predicates Ĥ_{i} of L_{c}:

1. G is a group;

2. the Ĥ_{i} ’s are subgroups;

3. the projection of X on the first k − 1 coordinates is equal to Y (where X is k-ary);

4. the cartesian product of X and Y is equal to Z;

5. the intersection of X and Y is equal to Z;

6. X is equal to the group Y up to a fixed permutation of coordinates;

7. the index of X in Y is equal to k;

8. the index of X in Y is greater or equal to k ;

9. the tuple c is in X;

Remark that the sentences of types 3, 4, 5 and 6 correspond to the equivalence sentences which occur in 𝔸.

Note that every subgroup G_{0} of G which contains all the constants, satisfies the axioms of types 2, 4, 5, 6, and 9. For each sentence Ψ_{X,Y,k} of types 7 or 8, a subgroup G_{0} satisfies Ψ_{X,Y,k} if and only if it contains at least k elements of Y which are in different cosets modulo X. Thus there exist finitely generated subgroups of G which satisfy the finite set of axioms of types 1, 2, 4, 5, 6, 7, 8 and 9.

To deal with axioms of type 3, we need to find finitely generated subgroups which are also “closed under projection” in the adequate sense. This is done in the two following claims. We say that a subset X of G is stable under quasi-endomorphisms if for each x ∈ X and each α ∈ 𝔽_{q}, the set {y ∈ G : (x, y) ∈ H_{α}} is a subset of X.

Claim. Let X be a definable subgroup of G^{k}. Then there exists a finite subset D_{X} of G such that, if G_{0} is any subgroup stable under quasi-endomorphisms which contains D_{X}, if π denotes the projection from G^{k} onto the first k − 1 coordinates, then π(X) ∩ G_{0}^{k−1} = π(X ∩ G_{0}^{k}).

Proof. Let l be the dimension (algebraic dimension = Morley rank) of X and (a_{1}, ..., a_{k}) a generic point of X, that is a point of dimension l. Then, there are two cases.

Either, a_{k} is independent of a_{1}, ..., a_{k−1}. It follows easily in this case that X = Y × G where Y = {(x_{1}, ..., x_{k−1}) : (x_{1}, ..., x_{k−1}, 0) ∈ X} and then for every subgroup G_{0}, π(X) ∩ G_{0}^{k−1} = π(X ∩ G_{0}^{k}).

Otherwise, by a permutation of coordinates we can assume that a_{l+1}, ..., a_{k} are algebraic over a_{1}, ..., a_{l}. (Note that then every generic of X satisfies this property.) For each j, l < j ≤ k, a_{j} ∈ acl(a_{1}, …, a_{l}); so (see Fact 2.1), there exist α_{j,1}, ..., α_{j,l} ∈ 𝔽_{q} and b_{j,1}, ..., b_{j,l} ∈ G such that a′_{j} = a_{j} − ∑_{1≤i≤l} b_{j,i} ∈ acl(∅) and for every i ≤ l, (a_{i}, b_{j,i}) ∈ H_{α_{j,i}}.

Let T be the subgroup of G^{k} of elements (x_{1}, ..., x_{k}) such that there exist y_{l+1}, ..., y_{k} with (x_{1}, ..., x_{l}, y_{l+1}, ..., y_{k}) ∈ X and for each j, l < j ≤ k, there exist y_{j,1}, ..., y_{j,l} with x_{j} = y_{j} − ∑_{i} y_{j,i} and i ≤ l, (x_{i}, y_{j,i}) ∈ H_{α_{j,i}}. Then (a_{1}, …, a_{l}, a′_{l+1}, …, a′_{k}) ∈ T. Now let T′ := {(x_{l+1}, ..., x_{k}) : (0, ..., 0, x_{l+1}, ..., x_{k}) ∈ T }.

We claim that T′ is finite and that T = G^{l} × T′. Since X is of dimension l, the group X′ := {(x_{l+1}, ..., x_{k}) : (0, ..., 0, x_{l+1}, ..., x_{k}) ∈ X} is finite. It follows that T′ is finite because the cokernels of the quasi-endomorphisms are finite. Let (x_{1}, ..., x_{l}) be a generic of G^{l} over (a_{1}, ..., a_{l}). By strong minimality, (x_{1}, ..., x_{l}) and (a_{1}, ..., a_{l}) have the same type over acl(∅). Since (a′_{l+1}, ..., a′_{k}) ∈ acl(∅), we have (x_{1}, ..., x_{l}, a′_{l+1}, ..., a′_{k}) ∈ T and so (x_{1} − a_{1}, ..., x_{l} − a_{l}, 0, .., 0) ∈ T. But (x_{1} − a_{1}, ..., x_{l} − a_{l}) is generic, so G^{l} × {0}^{k−l} ⊂ T and thus T = G^{l} × T′.

Now, let G_{0} be any subgroup of G stable under quasi-endomorphisms such that G_{0}^{k−l} contains T′. Then π(X) ∩ G_{0}^{k−1} = π(X ∩ G_{0}^{k}): indeed let (x_{1}, ..., x_{l}, y_{l+1}, ..., y_{k}) ∈ X be such that x_{1}, ..., x_{l} ∈ G_{0}, we are going to show that y_{k} ∈ G_{0}. For each j, l < j ≤ k, take y_{j,1}, ..., y_{j,l} such that for each i ≤ l, (x_{i}, y_{j,i}) ∈ H_{α_{j,i}}. Then, by stability of G_{0} under quasi-endomorphisms, y_{j,i} ∈ G_{0} for each j, l < j ≤ k, and each i ≤ l. For each j, l < j ≤ k, let z_{j} = y_{j,1} + ... + y_{j,l} and x_{j} = y_{j} − z_{j}. Then (x_{1}, ..., x_{l}, x_{l+1}, ..., x_{k}) ∈ T and as T = G^{l} × T′, (x_{l+1}, …, x_{k}) ∈ T′ ⊂ G_{0}^{k−l}. So, in particular, x_{k} ∈ G_{0} and thus y_{k} = x_{k} + z_{k} ∈ G_{0}.

Claim. For every finite subset A ⊂ G, there is a finite subgroup G_{0} of G, containing A, which is stable under quasi-endomorphisms.

Proof. Let A be a finite subset of G. For a subset X of G denote by \overline{X} the set ∪_{(x,α)∈X×𝔽_{q}}\{y ∈ G : (x, y) ∈ H_{α}\}. Note that \overline{X} is not necessarily stable under quasiendomorphisms (i.e. \overline{\overline{X}} is not necessarily equal to \overline{X}).

For each (α, β) ∈ 𝔽_{q}^{2}, let H_{α} ∘ H_{β} denote the subgroup of G^{2} defined by the formula

∃z ((x, z) ∈ H_{β} ∧ (z, y) ∈ H_{α}).

The quasi-endomorphism H_{αβ} is equal to the connected component of H_{α} ∘ H_{β}. Let X_{0} := {y ∈ G : (0, y) ∈ H_{α} ∘ H_{β} for some (α, β) ∈ 𝔽_{q}^{2}} and let A_{0} be the finite subgroup generated by A and \overline{X_{0}}. Let B be the set \overline{A_{0}}. We prove that B is stable under quasi-endomorphisms : let α ∈ 𝔽_{q}, x ∈ B and y ∈ G be such that (x, y) ∈ H_{α}. By definition of B, there exists β ∈ 𝔽_{q} and z ∈ A_{0} such that (z, x) ∈ H_{β}. So (z, y) ∈ H_{α}∘H_{β}. Let y′ ∈ G be such that (z, y′) ∈ H_{αβ}. Then y − y′ ∈ X_{0} since (0, y − y′) ∈ H_{α} ∘ H_{β}. Remark that if α = 0 then y = 0 and if β = 0 then y ∈ X_{0}. So assume that αβ ≠ 0. Let t ∈ G be such that (y − y′, t) ∈ H_{(αβ)^{−1}}. Then t ∈ A_{0} since y − y′ ∈ X_{0} and \overline{X_{0}} ⊆ A_{0}. Thus y ∈ B since (z + t, y′ + (y − y′)) ∈ H_{αβ} and z + t ∈ A_{0}. Now consider G_{0} the subgroup generated by B. Then G_{0} is also stable under quasi-endomorphisms since for each x_{1}, x_{2} ∈ G and each α ∈ 𝔽_{q},

{y ∈ G : (x_{1} + x_{2}, y) ∈ H_{α}} = {y_{1} ∈ G : (x_{1}, y_{1}) ∈ H_{α}} + {y_{2} ∈ G : (x_{2}, y_{2}) ∈ H_{α}}.

Now by the previous claims we can find a finite subgroup of G, which contains sufficiently many elements in different cosets for the axioms of type 7 or 8 to be satisfied, which is stable under quasi-endomorphisms and contains each D_{H_{i}}. Such a finite group is a model of 𝕋.

Corollary 3.17. 𝔸_{1} is a finite axiomatization of the complete theory 𝕋_{1}.

Proof. By lemma 3.15, it suffices to show that every model of 𝔸_{1} is infinite. Let 𝕊 be a model of 𝔸_{1}. We work in the structure 𝔾_{S} which is a model of 𝕋 by 3.14. For r ∈ K, let ϕ_{r} denote the corresponding pp-formula (over ∅). In 𝔾_{S}, the kernel and cokernel of ϕ_{r} are finite, hence, by Lemma 3.14, they must be contained in G ⊕ {0} This means that ϕ_{r} restricted to {0} ⊕ S is a well-defined map. Let s ∈ S \ {0}. For each r ∈ K, consider the unique (x_{r}, y_{r}) ∈ G_{S} such that ((0, s), (x_{r}, y_{r})) ∈ ϕ_{r}. Then, if r ≠ r′, y_{r} ≠ y_{r′}: indeed, if for r ≠ r′, y_{r} = y_{r′} then ((0, s), (x_{r} − x_{r′}, 0)) ∈ (ϕ_{r} − ϕ_{r′}), ((x_{r} − x_{r′}, 0), (0, s)) ∈ (ϕ_{r} − ϕ_{r′})^{−1} and hence s = 0.

By Proposition 2.4, 𝕋_{1} is interdefinable with the theory of non trivial K-vector spaces, where K is the ring of quasi-endomorphisms of 𝔾. By Corollary 3.17 and Lemma 3.2, the theory of K-vector spaces is finitely axiomatizable. By Proposition 3.13 we derive immediately:

Corollary 3.18. The division ring K of quasi-endomorphisms of 𝔾 is finitely presented as a ring.

3.4 Finitely axiomatizable strongly minimal groups

Theorem 3.19. Let G be a strongly minimal group. If T h(G) is finitely axiomatizable in a finite language L, then the quasi-endomorphisms ring of G is an infinite division ring which is finitely presented as a ring.

Proof. By [10], a finitely axiomatizable strongly minimal group must be locally modular, hence one-based.

Claim. If G is a strongly minimal locally modular group, and G′ is an expansion of G by a set C of constants, then G and G′ have the same quasi-endomorphisms ring.

Proof. . For A ⊂ G, let S be any connected A-definable subgroup of G × G in G′. Then S is also definable in G, over A ∪ C. As G is one-based, by Fact 3.5, S is definable over acl^{eq}(∅) in G. Conversely, any definable connected subgroup H in G remains connected in G′.

Recall the construction from Corollary 3.7: we add finitely many constants from M_{0}, the prime model of T h(G). Let G′ denote the expansion of G to the new language L′ = L ∪ {m_{0}, …, m_{n}}. Then G′ is interdefinable with some finitely axiomatizable abelian structure 𝔾 =< G, +, −, 0, H_{1}, …, H_{k}, m_{0}, …, m_{n} >. It follows that 𝔾 and G have the same quasi-endomorphisms ring, as a quasi-endomorphism is a definable connected subgroup of G × G. (Note that, as in 𝔾 every definable connected subgroup is defined over ∅ (Proposition 1.6), the same is true in G′. Hence in G every definable connected subgroup was already definable over {m_{0}, …, m_{n}}.)

By Corollary 3.18, the division ring of quasi-endomorphisms of 𝔾 is finitely presented as a ring. As remarked above, this is also the division ring of quasi-endomorphisms of G.

By Corollary 3.12, we derive immediately:

Corollary 3.20. Let G be a Morley Rank one group. If G is finitely axiomatizable then the quasi-endomorphism ring of its connected component is an infinite division ring which is finitely presented as a ring

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Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France. E-mail : blossier@math.univ-lyon1.fr

CNRS - Université Paris-Sud Orsay, Département de Mathématiques, Bat. 425, 91405 Orsay Cedex, France. E-mail : elisabeth.bouscaren@math.u-psud.fr