DMP IN STRONGLY MINIMAL SETS

ASSAF HASSON^{∗} AND EHUD HRUSHIVSKI^{†}

Abstract. We construct a strongly minimal set which is not a finite cover of one with DMP. We also prove that for a strongly minimal theory T generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.

Recall that a strongly minimal theory T has the Definable Multiplicity Property (DMP) if for all natural k, m and φ(x̄, b̄) of rank k, multiplicity m there exists a formula θ ∈ tp(b̄) such that for all b̄′ ⊧ θ, rk{φ(x̄, b̄′)} = k and mult{φ(x̄, b̄′)} = m. This definition was introduced in [2], where it was asked whether every strongly minimal set is a finite cover of one with DMP, i.e. whether for every strongly minimal T there exists a definable equivalence relation E with finite classes such that T/E has DMP.

In [4] Kikyo and Pillay prove that a strongly minimal T (with quantifier elimination) which is a finite cover of a theory with DMP has generic automorphisms (i.e. the theory T ∪{σ is an automorphism} has a model companion) iff T itself has DMP. They conjectured that the same is true of any strongly minimal theory.

In the first section of this paper we construct a strongly minimal set which is not a finite cover of one with DMP. In the second part we prove Kikyo and Pillay’s conjecture.

1. A strongly minimal set with no DMP

In this section we slightly change the construction of [3] to obtain:

Theorem 1.1. There exists a strongly minimal theory T such that for every definable infinite equivalence relation E with finite classes, T/E does not have DMP.

It seems that our construction methods can be generalized, but for the sake of simplicity and clarity we will stick to the simplest example. We will go very briefly through most of the definitions and standard statements, as they are practically similar to the corresponding ones in [3].

Let L := {S, R_{i}, a_{i}}_{i<ω} be a language consisting of one ternary relation S, countably many binary relations R_{i} and a set of constants {a_{i}}_{i<ω}. For any L structure A we denote Â := A \ {a_{i}}_{i<ω} and say that A is L−finite if Â is finite. For an L−finite structure A we denote R(A) := {z ∈ A^{3} : A ⊧ S(z)}, r(A) = |R(A)| and d_{0}(A) = |Â| − r(A). Let T_{∀} be the following theory:

(1) d_{0}(A) ≥ 0 for every L−finite structure A.

(2) R_{i}(z_{1}, z_{2}, z_{3}) ⇒ z_{1} = a_{i} ∧ S(a_{i}, z_{2}, z_{3}).

∗Supported in part by the EPSRC grant no. EP C52800X 1.

†Supported in part by Israel Science Foundation Grant no. 244/03.

Let C be the class of L−finite models of T_{∀}. For B_{1}, B_{2} ∈ C and A = B_{1} ∩ B_{2} we define the free join of B_{1} with B_{2} over A, denoted B_{1} ⊕_{A} B_{2} as the L−structure whose universe is B_{1}_{∐A}B_{2} and whose relations are precisely those appearing already in B_{1} or B_{2}. As usual we set for A ⊆ B ∈ C

d_{B}(A) = min{d_{0}(C) : A ⊆ C ⊆ B}

and A ≤ B if d_{B}(A) = d_{0}(A), in which case we also say that A is self-sufficient in B. If A ≤ B and d_{0}(B/A) := d_{0}(A ∪ B) − d_{0}(A) = 0 we say that B is prealgebraic over B. It is simply pre-algebraic if d_{0}(B′/A) > 0 for all A ⊊ B′ ⊊ B. If in addition A is minimal such that B/A is simply pre-algebraic we say that B/A is simply minimally pre-algebraic. It is standard to check that ≤ is transitive.

The following is easy and will be used without reference. The proof is left as an exercise:

Claim 1.2. Let F ⊆ A, B ⊆ N ⊧ T_{∀}. If B/F is minimally simply pre-algebraic, A ≤ N and B′ ∈ atp(B/F)^{N} then either B′ ∩ A = F or B′ ⊆ A.

Let B/A be minimally simply pre-algebraic. Let ā := ⋀_{i=1}^{3} π_{i}(R(B)) ∩ A for π_{i} the natural projection onto the i^{th} coordinate. Let ψ_{B/A}(y) ∈ atp_{S}(ā) isolate the type, and ϕ_{B/A}(x, y) isolate atp_{S}(B̂/ā). Let

A := {(B, A) : B/A is minimally simply algebraic}

.

We fix a function μ : A → ℕ with the following properties:

(1) If (B, A), (B′, A′) ∈ A and A′ ⊧ ψ_{B/A}, B′ ⊧ ϕ_{B/A}(x, A′) then μ(B, A) = μ(B′, A′).

(2) If (B, A) ∈ A and τ is a permutation on {1, …, |B|} fixing A setwise then μ(τ(B, A)) = μ(B, A).

(3) μ(B/A) ≥ max{d_{0}(A), 2}.

If B/A is simply pre-algebraic (but not necessarily minimally so) we may abuse the notation and write μ(B, A) meaning μ(B, F) for the unique F ⊆ A such that B/F is minimally simply pre-algebraic. In addition, by (2) above, there will not be any ambiguity in referring - as we will - to μ(B, A) without specifying an enumeration of B and A.

Remark 1.3. The requirement that μ(B/A) ≥ d_{0}(A) is not used explicitly in the text, but appears implicitly in the proof of Lemma 1.5 below. The requirement that μ(B/A) ≥ 2 does not appear in the original construction and will be only needed to obtain Lemma 1.15.

Definition 1.4. We will denote C_{μ} ⊆ C the sub-class of all L−finite M ⊧ T_{∀} in which there are at most μ(B/A) pairwise disjoint realizations of atp_{S}(B/A) for any fixed A ⊆ M and A ≤ B ∈ C minimally simply-algebraic.

The following is Lemma 3 of [3]. Both the statement and proof remain unaltered:

Lemma 1.5. Suppose that A, B_{1}, B_{2} ∈ C_{μ} are such that A ⊆ B_{1} and B_{1}/A is simply pre-algebraic. Assume for simplicity that A = B_{1} ∩ B_{2} and let E = B_{1} ⊕_{A} B_{2} then one of the following is true:

(1) E ∈ C_{μ}.

(2) There exists F ⊆ A such that B_{1}/F is minimally simply pre-algebraic and disjoint C_{1}, …, C_{r} ⊆ B_{2} such that atp_{S}(C_{i}/F) = atp_{S}(B_{1} \ F/F) and r = μ(B/F).

(3) There exists X ⊆ B_{2} such that X∩A ≰ X and B_{1} contains an S−isomorphic copy of X.

Concerning the adaptation of the proof to the present context we note only that since μ depends only on the S−structure of B/A the relations R_{i} can be ignored for the purpose of this lemma. The two differences with respect to the original proof are that our structures are not finite (since they all contain the constants) and that d_{0} is slightly changed. As both B_{1} and B_{2} are L−finite there is only a finite set of constants C which contributes to the S−type of B_{1} ∪ B_{2}, so we may restrict our attention to Ê := B̂_{1} ⊕_{ÂC} B̂_{2}, which eliminates the first difference with the original proof. Requiring that (keeping the notation in the statement of the lemma) C ∩ C_{i} = ∅ and that X ⊆ B̂_{2} eliminate the second difference.

In [3] self-sufficient amalgamation was an almost immediate corollary of (the analogue of) Lemma 1.5. In the present case this is clearly not enough, as the (quantifier free) S−type of any tuple does not isolate its (quantifier free) type. To obtain the self-sufficient amalgamation we add some more restrictions on our function μ and on the class C_{μ}.

Notation 1.6. Let B ⊇ A be such that |B \ A| < ∞. We set

c(B/A) := |S(B) ∩ (A × B^{2} \ A^{2})|

and

Co(B/A) := S(B) ∩ ((_{⋃i<ω}a_{i}) × B^{2} \ A^{2})
co(B/A) = |Co(B/A)| .

Remark 1.7. Let B/A be minimally simply algebraic and A′ ≡_{S} A. For any B′ such that atp_{S}(B′, A′) = atp_{S}(B, A) there are precisely 2^{co(B′/A′)} complete atomic L−types p_{i} ⊇ atp_{S}(B′, A′) ∪ atp_{L}(A′).

Proof. p_{i} as above will be determined by the R_{i} relations it implies. By T_{∀}(2) this consists in, for each s ∈ Co(B/A), deciding whether ⊧ R_{π_{1}(s)}(s) or ⊧ ¬R_{π_{1}(s)}(s) (π_{1} the projection on the first coordinate). Since the choices are independent the remark follows. □

Now we require that 2^{c(B/A)} | μ(B/A) and define Ĉ_{μ} ⊆ C_{μ} by requiring that for every simply minimally algebraic B/A and any A ⊆ M ∈ Ĉ_{μ} there are at most μ(B/A)/2^{co(B/A)} disjoint realizations of atp(B/A) in M. We can now prove:

Lemma 1.8 (Self-sufficient amalgamation). Suppose that A, A_{1}, A_{2} ∈ Ĉ_{μ}, A ≤ A_{i}. Then there exists E ∈ Ĉ_{μ} and embeddings f_{i} : A_{i} ↪ E such that f_{1} ↾_{A}= f_{2} ↾_{A} and f_{i}(A_{i}) ≤ E.

Proof. The proof is easily adapted from [3]. We give the details for completeness. We proceed by induction on |A_{1} − A| + |A_{2} − A|. Let d = d_{A_{i}}(A) = d_{0}(A). If there existed some A ⊊ X ⊊ A_{1} such that d_{0}(X) = d then X ≤ A_{i} and using the induction hypothesis twice we get the desired result.

If d_{0}(A_{1}) > d and there is no X as above, then there must be some b ∈ A_{1} \ A such that d(b/A) = 1 (in fact, any b will do). It is immediate that in that case A_{2} ∪ {b} ∈ Ĉ_{μ} and that A ∪ {b} ≤ A_{i}b, so that we can again proceed by induction.

It remains to deal with the case that A_{1}/A is simply pre-algebraic, so it is minimally simply pre-algebraic over some C ⊆ A. Since A ≤ A_{2} either (1) or (2) of Lemma 1.5 must hold (as (3) would contradict A ≤ A_{2}). If (2) is the case then atp_{S}(A_{1}/A) is realized μ(A_{1}/C) times disjointly in A_{2}. By Remark 1.7 this can only happen if among them are μ(A_{1}/C)/2^{co(A_{1}/C))} realizations of atp(A_{1}/A). Because A ≤ A_{2} the minimality of A_{1}/C implies that each such realization is either contained in A or freely joint with A over C. Since A_{1} ∈ Ĉ_{μ} the former case cannot happen. Hence there is a copy of A_{1} \ A in A_{2} freely joint with A over C and it is easy to check that we can identify A_{1} \ A with this copy of itself.

Finally, if (1) is the case then the free join of A_{1} with A_{2} over A contains at most μ(A_{1}/C) disjoint copies of atp_{S}(A_{1}/A). If no more than μ(A_{1}/C)/2^{co(A_{1}/C)} realize atp(A_{1}/C) we are done. Otherwise by the exact same argument, one of them must be disjoint form A and contained in A_{2} and we can identify A_{1} \ A with this image of itself. □

Corollary 1.9. There exists a (countable) structure U universal and homogeneous with respect to (Ĉ_{μ}, ≤), i.e. ∅ ≤ U and for every finite A ≤ U and A ≤ B ∈ Ĉ_{μ} (finite) there exists an embedding f : B ↪_{A} U such that f(B) ≤ U.

We are now ready to give the axiomatization of the strongly minimal set we are aiming at. To keep the axiomatization simple we will use the following notation:

For (B, A) ∈ A ϕ_{B/A}(x, y) be as in the definition of the function μ and set n = |B \ A| for m ∈ ℕ we define

θ_{B/A}^{m}(x_{1,1}, … x_{m,n}, A) := ⋀_{j_{1}≤j_{2},i_{1}≠i_{2}} x_{i_{1},j_{1}} ≠ x_{i_{2},j_{2}} ∧ ⋀_{i=1}^{m} ϕ_{B/A}(x̄_{i}, A)

where x̄_{i} := (x_{i,1}, …, x_{i,n}).

Define T_{μ} to be:

• T_{∀}

• For every model M and L-finite substructure B ⊆ M, B ∈ Ĉ_{μ}.

• For every model M, any L-finite B ⊆ M and B ≤ C ∈ Ĉ_{μ} such that C/B is minimally simply pre-algebraic and such that X ∩ B ≤ X whenever X ⊂ M realizes an atomic type realized in C there are μ(C/B) solutions of atp_{S}(C/B) which are pairwise disjoint and any other solution of atp_{S}(C/B) meets at least one of them. Formally, if the assumptions of the statement hold then:

M ⊧ (∃x̄_{1}, … x̄_{r})θ_{C/B}^{m}(x̄_{1}, … x̄_{r}, B) ∧ (¬∃x̄_{0}, … x̄_{r})θ_{C/B}^{m+1}(x̄_{0}, … x̄_{r}, B)

for r = μ(C/B).

• For every k ∈ ℕ, every fixed set of constants ā := {a_{i_{1}}, …, a_{i_{k}}} every M, B, C as above and such that ā ⊆ Co(C/B) given μ(C/B) solutions of atp_{S}(C/B) as above there are precisely μ(C/B)/2^{|ā|} among them satisfying each of the 2^{|ā|} completions of atp_{S}(C/B) to a complete atomic type in S ∪ {ā ∪ R_{i_{1}}, …, R_{i_{k}}}.

The last axiom scheme in the definition of T_{μ} introduces the main novelty of the present construction with respect to the one in [3]. In the original construction, if B/A is minimally simply algebraic its atomic type is isolated uniformly in A. This is not the case in the present situation. A single axiom of this last axiom scheme (for one q.f.-S-type of B) cannot cover all possible L-types of B. The new point is that other axioms, referring to specializations of B also, precisely fill in the gaps left by the axiom referring to B itself. This relies heavily on the fact that we have a precise count of the number of disjoint realizations of ϕ_{C/B}(x, y) for all y ⊧ ψ_{C/B}(y), with the count independent of y.

Claim 1.10. T_{μ} is first order.

Proof. d_{0} ≥ 0 can be stated as:

For every finite subset C of constants and any disjunction of atomic S-formulas ϕ(x̄, ȳ) with more than |x̄| distinct literals of the form S(x_{i_{1}}, x_{i_{2}}, x_{i_{3}}) with 1 ≤ i_{1} ≤ i_{2} ≤ i_{3} ≤ |x|, an axiom (⋀_{i≠j} x_{i} ≠ x_{j}) → ¬ϕ(x̄, c̄) for any enumeration c̄ of C.

So T_{∀} is first order. The second axiom scheme (modulo T_{∀}) simply bounds the number of solutions of certain equations, so is clearly first order. By condition (1) in the definition of μ the last two axiom schemes are given in such a way that every instance of them is already stated in first order way.

□

Remark 1.11. Note that (once the consistency of T_{μ} is verified, if B ≤ M ⊧ T_{μ} and (C, B) ∈ A then we can strengthen the conclusion in the third axiom scheme. In that case we can actually state that there are μ(C/B) solutions of atp_{S}(C/B) in M which are all pairwise disjoint and such that any other solution of atp_{S}(C/B) is a permutation of one of them.

With this in hand the following claim is an immediate corollary of self-sufficient amalgamation (see e.g. Section 3 of [3] for more details) once we show that T_{μ} is consistent:

Claim 1.12. Let M_{1}, M_{2} be saturated models of T_{μ} of the same cardinality and A_{i} ≤ M_{i}, |A_{i}| < |M_{i}| then any isomorphism f : A_{1} → A_{2} extends to an isomorphism of M_{1} with M_{2}.

We can now prove:

Lemma 1.13. T_{μ} is consistent, complete and strongly minimal.

Proof. To check that T_{μ} is consistent we only have to show that U of Corollary 1.9 satisfies the axioms. By self-sufficient amalgamation everything is clear, except possibly the the case where B ⊆ U is not self-sufficient and we are given B ≤ C to embed into U. Let B̄ := ⋂{B′ : B ⊆ B ≤ U} and C̄ = C ⊕_{B} B̄. If C̄ ∈ Ĉ_{μ} and there exists X ⊆ C̄ (realized in U) such that X ∩ B̄ ≰ X then X ∩ B ≰ X ∩ C (because for all c ∈ C̄ \ B̄ we have d_{0}(c/B̄) = d_{0}(c/B)), and there is nothing to do. If for all X ⊆ C̄ realized in U we have X ∩ B̄ ≤ X we can easily conclude by self-sufficient amalgamation.

By Claim 1.12 to show that T_{μ} is complete we only have to make sure that the atomic type of the constants is determined and that ⋃ a_{i} ≤ U. This follows from the axiom that d_{0}(A) ≥ 0 for all finite A ⊆ U.

It remains to check that T_{μ} is strongly minimal. We have to show that if M ⊧ T is saturated and A ⊆ M is small then there exists a unique 1−type with infinite orbit under Aut(M/A). We may assume that A ≤ M (since defining Ā as above, Ā ⊆ acl(A)). For b_{1}, b_{2} ∈ M such that d(b_{i}/M) = 1 we have by Claim 1.12 that tp(b_{1}/A) = tp(b_{2}/A) so we only have to show that if d(b/A) = 0 then b ∈ acl(A). Let b̄ minimal such that b ∈ b̄ and d_{0}(b̄/A) = 0, then b̄/F is minimally simply prealgebraic for some finite F ⊆ A, and therefore has at most finitely many disjoint realizations in M. By the minimality of b̄/A and the assumption that A ≤ M any two realizations of atp(b̄/A) must be disjoint, which implies that atp(b̄/a) and therefore tp(b̄/A) has only finitely many realizations in M, as required. □

Before we proceed to to showing that T_{μ} has the properties we want some comments on the construction itself are in place. First, it follows immediately from the axiomatization that T_{μ}|_{S} corresponds to the construction (of [3]) with (S, d_{0}, μ). Hence T_{μ}|_{S} has DMP, whereas T_{μ} clearly doesn’t. Since in both theories acl(A) = {a : d(a/A) = 0} it follows T_{μ} is a non-trivial acl-preserving expansion of T_{mu}|_{S}.

We now turn to proving satisfies the main property we are interested in:

Lemma 1.14. T_{μ} is not a finite cover of a theory with DMP (in particular T_{μ} does not have DMP).

Proof. Let E be any definable equivalence relation with finite classes. We show that the induced structure M^{∗} = M/E does not have DMP. Denote S^{∗} = S /E and R_{i}^{∗} = R_{i} /E. For a ∈ M denote S_{a} := S(a, x_{1}, x_{2}). We show that R_{i}^{∗} splits S_{a_{i}}^{∗} into two infinite sets for almost all i. First, it is clear that rk{S_{a_{i}}^{∗} ∩ R_{i}^{∗}} ≥ 1 (for if a_{i}^{∗} = a_{i} /E then for all generic (b_{1}, b_{2}) ∈ S_{a_{i}}^{∗} ∩R_{i}, M_{2}^{∗} ⊧ S^{∗}(a_{i}^{∗}, b_{1}^{∗}, b_{2}^{∗})∧R_{i}^{∗}(b_{1}^{∗}, b_{2}^{∗})). So it will be enough to show that if b is a generic element (over C, the set of parameters used to define E) and (b_{1}, b_{2}) ∈ S_{b} is a generic element in the corresponding fiber then M ⊧ S(b, b′_{1}, b′_{2}) for b′_{i}Eb_{i} iff b_{1} = b′_{1} and b_{2} = b′_{2}. Assume not then (assuming without loss that C ≤ M) we get that d(b′_{1}/Cb_{1}) = 0 (for E has finite classes and algebraic closure in M_{2} corresponds to the relation d(a/B) = 0), and let B ⊇ b′_{1} be a minimal finite set such that d_{0}(B/Cb_{1}) = 0. Because, as above, b_{2} | ⌣b_{1}necessarily b_{2}, b′_{2} ∉ B and therefore d_{0}(B/Cb_{1}, b′_{2}, b) < 0. We show that this cannot be:

d_{0}(B/Cb_{1}, b′_{2}, b) = d_{0}(Bb_{1}/Cb′_{2}, b)

so d_{0}(Bb_{1}/Cb′_{2}, b) < 0, and so d_{0}(Bb_{1}b′_{2}/Cb) ≤ 0 but this implies that b_{1} ∈ acl(b), a contradiction.

But for generic a^{∗} the set S_{a^{∗}}^{∗} is strongly minimal so no DMP follows by strong minimality of T_{μ}^{∗}. □

More generally we can prove:

Lemma 1.15. T_{μ} has no definable (with parameters) equivalence relations (on singletons) with infinitely many classes with more than one point.

Proof. Assume that E is a definable equivalence relation contradicting the statement of the lemma. Let C ≤ M be such that E is defined over C, and choose b any element of an E-class generic over C. Note that for any b′ ⊧ E(x, b) we have b′ ∈ acl(Cb). It follows that d(b′/Cb) = 0. As b is generic over C and E is non-trivial there exists b′ ≠ b as above, and for any such b′ there exists B′ ∋ b′ such that B′/Cb is simply pre-algebraic. Since μ(B′/Cb) ≥ 2 there exist disjoint B_{1}, B_{2} such that atp(B′/Cb) = atp(B_{i}/Cb) (this uses also the third bullet of the axiomatization of T_{μ} which assures that since Cb is self-sufficient the number of disjoint solutions of atp(B′/F) is precisely μ(B′/F) for F ⊆ Cb such that B′/F is minimally simply pre-algebraic). Since b is generic over C and C ≤ M also Cb ≤ M. Hence tp(B_{1}/Cb) = tp(B_{2}/Cb). In particular, there are b_{i} ∈ B_{i} such that ⊧ E(b_{i}, b).

By transitivity of E it follows also that d(b_{2}/Cb_{1}) = 0 and we choose a minimal set B_{1,2} witnessing it. We show that d_{0}(B_{1}B_{2}B_{1,2}b/C) = 0, which will contradict the fact that b ∉ acl(C). By the minimality of B_{1,2} and the fact that b_{2} ∈ B_{1,2} ∩ B_{2} it follows that d_{0}(B_{1,2}/CB_{1}B_{2}) ≤ 0 and equality holds iff B_{1,2} ⊆ B_{1}B_{2}. If d_{0}(B_{1,2}/CB_{1}B_{2}) < 0 then d_{0}(B_{1}B_{2}B_{1,2}b/C) = 0, and we have nothing to do. Otherwise it follows that d_{0}(B_{2}/CB_{1}b) < 0, and the contradiction is obtained in a similar way. □

Examining the proof of the last lemma we see that in precisely the same way can prove the stronger result:

Lemma 1.16. Let D ⊧ T_{μ} and S ⊆ D^{n} a quantifier free definable strongly minimal set. If E is a definable equivalence relation on S with finite classes and b̄_{1} ∈ S is generic over all the data then there exists b ∈ b̄_{1} such b ∈ b̄_{2} for all b̄_{2} ⊧ E(b̄_{1}, y) and b is generic over all the data.

In the last lemma b̄_{1} ∩ b̄_{2} means the intersection of the ranges of the tuples b̄_{1}, b̄_{2}. Also note that if C ≤ D is such that S, E are definable over C then, as S is quantifier free definable, for any b̄ ∈ S generic over C we have that Cb̄ ≤ D. Hence the proof of Lemma 1.15 goes through unaltered if instead of taking a generic b ∈ D (as we did there) we take b̄ ∈ S generic.

The following remark shows that the assumption that S in the previous lemma was quantifier free definable does not really harm the generality.

Remark 1.17. The reason for requiring that S be quantifier free definable is to assure that for C ≤ D and generic b ⊧ S we have Cb ≤ D. Note, however, that if S ⊆ D^{n} is any strongly minimal subset and E an equivalence relation with finite classes on S then there exists m ≥ 0, a quantifier-free definable strongly minimal set S′ ⊆ D^{n+m} and an equivalence relation E′ with finite classes on S′ such that π(S′) = S and E′ is the pre-image of E under π, the projection on the first n coordinates.

Combining Lemma 1.16 with the last remark we get:

Lemma 1.18. T_{μ} is not bi-interpretable with any strongly minimal theory with DMP.

Proof. We easily reduce to proving that if S is a strongly minimal set definable in T_{μ} then for every infinite equivalence relation E with finite classes S/E does not have DMP. By the last remark this reduces to proving the above for S that is quantifier free definable. Hence, we may take S, E as in lemma 1.16, and adding constants to the language we may also assume them to be ∅-definable. Assume first that if b̄_{1} ∈ S is generic and ⊧ E(b̄_{1}, b̄_{2}) then their ranges differ (i.e. there exists b ∈ b̄_{1} \ b̄_{2}). Let b ∈ b̄_{1} ∩ b̄_{2} be as provided by Lemma 1.16. And assume that b is the i_{1}^{th} element of b̄_{1} and the i_{2}^{th} element of b̄_{2}. Then b̄_{i} is pre-algebraic over b. If φ_{i_{j}} is the formula isolating atp(b̄_{i}/b) then it also isolates (in its rank) tp(b̄_{i}/b̄_{3−i}) whence for all b̄′_{i} ⊧ φ_{i_{j}} we have ⊧ E(b̄′_{i}, b_{3−i}). We obtained: if b̄_{1} ∩ b̄_{2} = {b} and b is the i_{1}^{th} element of b̄_{1} and the i_{2}^{th} element of b̄_{2} then E(b̄_{1}, b̄_{2}) iff ⋀_{1,2} b̄_{i} ⊧ φ_{i}(x, b). Repeating the same process for all possible i, j we get a complete description of E. We leave it as an exercise to check that in that case S/E has no DMP.

We now have to check the case that for some b̄_{1} ∈ S generic and b̄_{2} ⊧ E(b̄_{1}, y) the ranges of the two tuples agree. Let τ be a permutation such that τ(b̄_{1}) = b̄_{2} then E is clearly preserved under the action of the group generated by τ. So let G be the maximal permutation group which acts on the classes of E. Let E_{G} be the equivalence relation E_{G}(b̄_{1}, b̄_{2}) ⟺ b̄_{1} = τb̄_{2} for some τ ∈ G. If we quotient out by E_{G} we are in a situation very similar to the one treated above, and the claim follows, more or less as above. □

Remark 1.19. Note that the theory T_{μ}^{S} := T_{μ}|_{S} is precisely the theory constructed in [3] (for a μ-function satisfying some additional requirements). Therefore T_{μ}^{S} is strongly minimal with DMP and

a ∈ acl_{T_{μ}^{S}}(A) ⟺ d(a/A) = 0.

Since d(a/A) = 0 depends on S alone in T_{μ} it follows that the latter is a non-trivial acl-preserving expansion of T_{μ}^{S}.

It may be worth mentioning that the above example indeed does not have DMP but every reduct thereof to a finite sub-language does. It seems, however, that this can be remedied rather easily. Instead of requiring that μ depends only on L|_{S} we allow μ to depend on R_{i} as well (subject to the obvious requirement that if A_{a}, … A_{k} are all the possible specialization of some L̂-finite structure A, B/A is minimally simply pre-algebraic and atp_{S}(B_{i}/A_{i}) = atp(B/A) then ∑_{i} μ(B_{i}/A_{i}) = μ(B/A)). If μ is chosen generically enough, then R_{i} will be reconstructible from T_{μ}|_{S}. In the following paragraphs we sketch a different construction which gives the same result. Our construction is a variant of the construction of [1] in which we integrate the ideas used above.

We work in a two sorted language, the sorts being S, E in a language with two function symbols L = {f, π}. Let T be the theory asserting that π : S → E has infinite fibers; f : S → S preserves the fibers of π and is generically 1:1, but for each prime p ∈ ℕ there is a unique special p_{1}-fiber on which either f^{p}(x) = x or f^{k}(x) ≠ f^{j}(x) for all j ≠ k ∈ ℤ, and both cases occur infinitely often. For convenience we will add constants s_{p} ∈ E such that π^{−1}(s_{p}) is the special p-class. These constants will be used just to simplify the construction, then they will be discarded. We leave it as a simple exercise to check that T is a complete ω-stable theory of rank 2 with weak DMP (see [1]) for the definition).

We add a new sort D and a binary relation R ⊆ S × D. Define for a finitely generated L ∪ {R} structure A

d_{0}(A) = MR(A \ D) + |D ∩ A| − |R(A)|.

We define A ≤ B as usual, with the additional requirement that A is a substructure. We say that B/A pre-algebraic if:

(1) B/A is finitely generated.

(2) A ≤ B and d_{0}(B/A) = 0. If in addition

(3) B/A is minimal such that the above hold we say that B/A is simply prealgebraic.

(4) If also A is minimal then B/A is minimally simply pre-algebraic.

The following is easy:

Claim 1.20. If B/A is simply pre-algebraic there exist finite b ⊆ B and a ⊆ A such that:

• b generates B (over A).

• b, a are closed under π_{i}.

• For any d ∈ B if R(d, c) for some c ∈ A then c ∈ a and d ∈ B.
• If c := π_{1}(d) ∈ A for some b ∈ B then c ∈ a.

If we require that b is minimal satisfying the above then it is unique, and similarly for a.

We leave the proof as an exercise.

Definition 1.21. A finite tuple b/A is (simply) pre-algebraic if the structure it generates is pre-algebraic and b satisfies the conclusions of the last claim.

Remark 1.22. Let b/A be simply pre-algebraic. There exists a unique a ⊆ A such that b/a is minimally simply pre-algebraic. The atomic type of b/a is isolated. We will identify minimally simply-pre algebraic tuples with the (atomic) formulas isolating them. Note also that if b/a is minimally simply pre-algebraic isolated by φ(x, a) there exist a formula θ ∈ atp(a) such that for all a′ ⊧ θ if b′ ⊧ φ(x, a′) then b′/a′ is minimally simply pre-algebraic.

Note that the free amalgam of L ∪ {D} structures can be defined precisely as above.

The following is an immediate adaptation of the Algebraic Amalgamation Lemma of [3] (see also above) to the present context. We state it in a slightly different way.

Lemma 1.23. Let A = B_{1} ∩B_{2} be L∪{D} structures, A ≤ B_{1} and E = B_{1} ⊕_{A} B_{2}. Let d ⊆ E and φ(x, d) be such that a ⊧ φ(x, d) implies that a/d is minimally simply algebraic. There exists a natural number N depending on φ (but not on B_{1}, B_{2}) such that for any d ⊆ E one of the following holds:

(1) d ⊆ B_{2}.

(2) There exists a set X ⊆ B_{2} such that X ∩ A ≰ X and B_{1} contains an isomorphic copy of X.

(3) There are at most N disjoint realizations of φ(x, d) in E.

We are now in a situation very much like the the one we encountered in the first construction of this paper. To see this, note that if b̄/ā is minimally simply pre-algebraic, θ(ā) isolates the atomic type of ā/∅ and φ(x̄, ȳ) isolates the atomic type of b̄/ā then for any ā′ ⊧ θ there are at most 2^{|b̄|} different atomic types in φ(x̄, ā′) which are not outright algebraic (i.e. of negative relative pre-dimension). Ultimately, this follows from the fact that T has weak DMP, but this can be easily verified directly, by checking that for any such ā′, b̄′ ⊧ φ(x̄, ā′), and any b ∈ b̄′ such that π(b′) = s_{p} ∈ ā′, deciding whether or not f^{p}(b′) = b′ or not determines a complete non-algebraic atomic type. Moreover, all possible such completions are admissible. This allows us to define c(b̄/ā) exactly as above.

Finally, it is easy to check that if b̄/ā is minimally simply pre-algebraic, and s_{p} ∈ ā then there a minimally simply pre-algebraic couple b̄′/ā′ such that s_{p} ∉ ā′ and b̄/ā is a specialization of b̄′/ā′. Call a minimally simply pre-algebraic couple b̄/ā stationary if s_{p} ∉ ā of any prime p ∈ ℕ. Call an atomic φ(x̄, ȳ) stationary if it is minimally simply pre-algebraic and is satisfied by some stationary couple. It is standard book-keeping to enumerate (in a non-redundant way) stationary formulas so that every stationary couple satisfies precisely one of them. From this point on, we can proceed in choosing μ satisfying precisely the same requirements as we did above, and proceed with the construction accordingly, using Lemma 1.23.

2. DMP and generic automorphisms

Let T be any strongly minimal theory with QE, and T_{σ} := T ∪ {σ is an automorphism}. Denote by T A the model companion of T_{σ} if it exists. In [4] the following is shown:

Proposition 2.1. Let T be a strongly minimal theory, M ⊧ T a saturated model. Assume that for some definable equivalence relation E with finite classes T h(M/E) has DMP. Then T A exists iff already T has DMP.

In their paper Kikyo and Pillay conjecture that indeed this is true in general, i.e. that for strongly minimal theories DMP is equivalent to the existence of T A, and remark that if it were true that every strongly minimal set is a finite cover of a strongly minimal set with DMP then the conjecture will follow. We show that the conjecture is true in general. We will use:

Lemma 2.2. Let G be a finite group, H < G a subgroup such that H represents every conjugacy class of G (i.e. for every τ ∈ G there exists π ∈ G such that πτπ^{−1} ∈ H) then H = G.

Proof. By assumption G = ⋃ H^{G}. Now we compute:

|G| = | ⋃ H^{G}| ≤ (|H| − 1) |G/H| + 1 = |G| − |G/H| + 1 ≤ |G|

and equality holds iff H = G. □

We will use the following characterization of DMP which already appears (somewhat implicitly) in [2]:

Lemma 2.3. Let T be a strongly minimal theory, then the TFAE:

(1) T has DMP.

(2) Let P := P (a) be a strongly minimal set, G a finite group acting regularly and definably on P, D a strongly minimal set definable over ∅ and f : P → D a definable map such that the fibers of f are the orbits of G (except finitely many). Assume that:

(*) For almost all G−orbits on P the opposite action of G on the orbit is automorphic

Then there exists a formula φ ∈ tp(a) such that for all a′ ⊧ φ (*) is true of P (a′).

Proof. If T does not have DMP then (2) of the lemma does not hold: from [2] we know that NDMP is witnessed by a strongly minimal set P (a) = ψ(x, a). Assume wlog that for a generic b ⊧ ψ(x, a), b = (b_{1}, …, b_{n}), b_{1}/a is strongly minimal and b ∈ acl(ab_{1}). Let π_{1} : P (a) → D (where D is x = x) be the projection onto the first coordinate. There is no harm in assuming that for all b^{1}, b^{2} ∈ P (a) if π_{1}(b^{1}) = π_{2}(b^{2}) := b_{1} then there exists some σ ∈ Sym(n) such that σ(1) = 1 and σ(b^{1}) := (b_{σ(1)}^{1}, …, b_{σ(n)}^{1}) = b_{2}. Furthermore, we may assume that for almost all d ∈ D, |π^{−1}(d)| = m for some constant m ∈ ℕ. Finally, choose d ∈ D generic over a and let

G = {σ ∈ Sym(n − 1) : (d, b_{2}, …, b_{n}) ∈ P (a) ⇒ (d, b_{σ(2)}, …, b_{σ(n)}) ∈ P (a)}

Since we have chosen d generic over a it follows that G acts regularly (and obviously definably) on almost all the fibers of π_{1}. Note also that it cannot be that |G| = 1, as this would contradict the assumption that P (a) is a counter example to DMP. It is now immediate to check that by NDMP this is a counter example for (2).

For the other direction, assume that T has DMP and that P (a), D, f and G are as in (*). We have to show that the conclusion of (2) holds. Choose θ ∈ tp(a) such that for all a′ ⊧ θ, P (a′) is strongly minimal and we may further restrict θ so that for all (b_{1}, …, b_{n}) ∈ P (a′), b_{1} is interalgebraic with b_{i} for all i and that (b_{σ(1)}, …, b_{σ(n)}) ∈ P (a′) ⟺ σ ∈ G (we may of course think of G as a permutation group, in accordance to its action on a generic fiber of P (a)), then by strong minimality of P (a′) the opposite action of G on the fibers of P (a′) is automorphic almost everywhere, and we are done. □

Now we are ready to prove:

Theorem 2.4. Let T be a strongly minimal theory, then T A exists iff T has DMP.

Proof. The proof that if T has DMP then T A exists appears in [4]. For the other direction suppose that T A exists. Let P (a) be strongly minimal, we have to find θ ∈ tp(a) such that a′ ⊧ θ ⇒ P (a′) is strongly minimal. As in the direction ”⇒” of the previous lemma we can define a map π_{1} : P (a) → D with finite fibers, and assume that any two elements in a generic fiber differ only by a permutation, and let G be the corresponding permutation group. Let θ ∈ tp(a) be such that:

• For all a′ ⊧ θ, P (a′) has rank 1;

• If (b_{1}, …, b_{n}) ∈ P (a′) then for all i, b_{i} is interalgebraic with b_{1} and

• For all τ ∈ Sym(n − 1), (b_{1}, b_{τ(2)}, …, b_{τ(n)}) ∈ P (a′) ⟺ τ ∈ G. In other words, G acts regularly (and, of course, definably) on all the fibers of π_{1}(a′) for all a′ ⊧ θ.

Let M ⊧ T A and let a′ ∈ θ(M) be such that σ fixes a′ (pointwise). For each τ ∈ G there exists b_{τ} := (b_{1}, …, b_{n}) ∈ P (a′)^{M} such that σ(b_{1}) = b_{1} and σ(b_{i}) = b_{τ(i)} for i > 1 (b_{τ} exists since a′ ⊧ θ implying that G acts automorphically on the fibers of π_{1} and using the fact that M is existentially closed we can find b_{τ} as required).

Finally we show that this θ satisfies the requirements. Let a′ ⊧ θ^{M} for a large saturated M ⊧ T. Expand M to M′ ⊧ T A in such a way as to have σ ↾_{acl_{T} (a′)}= Id. Let τ ∈ G and take b_{τ} := (b_{1}, …, b_{n}) ∈ P (a′) as promised in the previous paragraph (i.e. σ(b_{1}) = b_{1} and σ(b_{i}) = b_{τ(i)} for i > 1), and let (d_{1}, …, d_{n}) be any generic of P (a′). Then tp(d_{1}/ acl_{T} (a′)) = tp(b_{1}/ acl_{T} (a′)) (both being generics of D) so there is some α ∈ Aut(M/ acl(a′)) with α(b_{1}) = α(b_{2}) then α^{−1}τa acts automorphically on the fiber above d_{1}. since α was an automorphism, this implies that the subgroup of G^{op} acting by automorphisms on the fiber of d_{1} meets every conjugacy class of G, so by what we have already shown, this implies that G^{op} acts by automorphism on the fiber of d_{1}. By the previous lemma this implies that T has DMP. □

Acknowledgment: The authors wish to thank Frank Wagner for pointing out a gap in the original proof of lemma 2.2 and Gregory Cherlin for supplying them with its elementary proof given in the text.

References

[1] A. Hasson. Interpreting structures of finite Morley Rank in strongly minimal structures. Preprint (2003).

[2] E. Hrushovski. Strongly minimal expansions of algebraically closed fields. Israel Journal of Mathematics 79, 129–151 (1992).

[3] E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic 62, 147–166 (1993).

[4] H. Kikyo and A. Pillay. The definable multiplicity property and generic automorphisms. Annals of Pure and Applied Logic 106, 263–273 (2000).

∗Mathematical Institute, Oxford University, Oxford, UK

E-mail address: hasson@maths.ox.ac.uk
URL: http://www.maths.ox.ac.uk/~hasson/

†Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

E-mail address: ehud@math.huji.ac.il