PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0

AROUND PODEWSKI’S CONJECTURE

KRZYSZTOF KRUPIŃSKI, PREDRAG TANOVIĆ, AND FRANK O. WAGNER

Abstract. A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski’s conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case).

We also study minimal groups with a (partial) order, and give a complete classification of almost linear minimal groups as certain valued groups.

1. Introduction

Recall that an infinite first-order structure is minimal if every definable (with parameters) subset is either finite or co-finite (of finite complement). Minimal pure groups were classified by Reineke [7]; they are either abelian divisible with only finitely many elements of any given finite order, or elementary abelian of prime exponent. As for minimal fields, it is well-known that every algebraically closed pure field is minimal; the converse was predicted by Podewski [6] forty years ago:

Conjecture 1. A minimal field is algebraically closed.

It was shown by the third author in positive characteristic [10], but the characteristic zero case remains one of the oldest unsolved problems in algebraic model theory.

We shall say that a structure is ordered if it has a definable strict partial order on singletons which is not definable from equality. Otherwise the structure is unordered. Please note that in this paper ordered fields or groups are ordered structures in the above sense rather than in the usual algebraic sense.

Lemma 1.1. A minimal ordered structure M has an infinite chain.

Proof. If there is a chain of order type ω^{∗}, the reverse order of ω, we are done. So assume that there is no such chain; we shall construct inductively a chain of order type ω.

Suppose the set X of minimal elements is co-finite. Then the elements of X are incomparable. By minimality, for each y ∈ M \ X the sets of elements less than y,

\begin{matrix}c \\ char∞∋ \end{matrix}1997 American Mathematical Society

Received by the editors 4 August 2012.

2010 Mathematics Subject Classification. Primary 03C60, Secondary 12L12, 20A15, 03C45 .

Key words and phrases. Podewski’s Conjecture; minimal field; minimal group; valued group.

Research supported by the Polish Government grant N N201 545938.

Supported by the Ministry of Education and Science of Serbia.

Partially supported by the Agence Nationale de la Recherche under project ANR-09-BLAN-0047 Modig.

bigger than y and incomparable to y are definable from equality. Hence the partial order is definable from equality, a contradiction.

Therefore the set of minimal elements is finite. If it is empty, there is a chain of order type ω^{∗}, a contradiction. Hence for some minimal element x_{0} the set X_{0} = {y : x_{0} < y} must be infinite, and hence co-finite by minimality. But if we can define the order on some co-finite set from equality, then, by minimality, we can define the order on M from equality. By induction, we obtain an infinite increasing chain. □

The minimal total orders are just (ω + n, <), ((ω + n)^{∗}, <) and (ω + ω^{∗}, <). Most of the known non-linear, ordered, minimal structures derive from them by replacing elements by sufficiently large finite antichains and then adding a finite set arbitrarily. For example, consider {(n, m) ∈ ω × ω | m ≤ n} and order it by (n, m) < (n′, m′) iff n < n′. Here, the order is not far from being linear, namely the incomparability relation (defined by x ∼ y iff ¬(x < y ∨ y < x)) is transitive and hence an equivalence relation; after factoring it out we end up with (ω, <). The minimal structures in which there is a definable order with an infinite chain such that incomparability is transitive will be called almost linear (see Definition 2 and Remark 2.1).

In Section 3, we prove Conjecture 1 for unordered fields:

Theorem 1. A minimal unordered field is algebraically closed.

Thus Podewski’s conjecture is reduced to the ordered case:

Conjecture 2. There is no minimal ordered field of characteristic zero.

We study minimal ordered groups in Section 4 and show:

Theorem 2. An almost linear minimal group G is either elementary abelian of exponent p or a finite sum of Prüfer p-groups for a fixed prime p. In particular, it is a torsion group.

This implies immediately:

Theorem 3. There is no almost linear minimal field.

It is thus natural to ask:

Question 1. Is every minimal ordered group a torsion group?

Theorem 3 implies that a possible counterexample to Conjecture 2 would have to be a field which is not almost linear. It is hard to believe that such structures exist. In particular, all known examples of minimal ordered structures are almost linear.

Question 2. Does there exist a minimal ordered structure [group] which is not almost linear?

Note that the analogue of Theorem 3 for quasi-minimal fields (uncountable fields whose definable subsets are countable or co-countable) is false: There is an almost linear quasi-minimal field [5, Example 5.1].

Finally, in Section 5 we classify almost linear minimal groups, showing in particular that all cases in the conclusion of Theorem 2 can be realized (so the analogue of Theorem 3 for groups is false).

2. Minimal structures with definable generic type

Let M be a minimal structure. The unique non-algebraic type p ∈ S_{1}(M) will be called the generic type of M. In this section, we are interested in minimal structures whose generic type is definable. This class is interesting because of the next lemma, noticed by A. Pillay.

Lemma 2.1. The generic type p of a minimal group G is its unique generic type in the sense of left [and right] translates, i.e. a formula ϕ(x) (with parameters from M) belongs to p if and only if finitely many left [right] translates of ϕ(G) cover G. In fact, ϕ(x) ∈ p if and only if two left [right] translates of ϕ(G) cover G. This characterization of p implies that p is definable over ∅.

Proof. If finitely many translates of ϕ(G) cover G, then ϕ(G) is infinite, hence ϕ(x) ∈ p. For the converse, suppose ϕ(x) ∈ p. Then G \ ϕ(G) is a finite set {g_{1}, …, g_{n}}. Since the sets g_{1}ϕ(G)^{−1}, …, g_{n}ϕ(G)^{−1} are co-finite, their intersection is non-empty, so it contains an element g. It is clear now that G = ϕ(G)∪gϕ(G). □

For the remainder of this section M will be a minimal structure whose generic type p ∈ S_{1}(M) is definable over ∅. Let M̄ ≻ M be a monster model. Then p has a unique global heir p̄ ∈ S_{1}(M̄), which is defined by the same defining scheme as p. For C ⊆ M̄ a generic Morley sequence is a sequence (a_{i} : i ∈ κ) such that a_{i} ⊧ p̄|Ca_{<i} for all i ∈ κ. Each generic Morley sequence over C is indiscernible over C, and the type over C of such a sequence of a fixed length κ does not depend on its choice. Recall that for a formula φ(x; \overline{y}) a φ(x; \overline{y})-definition of p is denoted by d_{p}xφ(x, \overline{y}).

Proposition 2.1. For any A ⊆ M̄ put

cl(A) = {x ∈ M̄ : tp(x/A) is non-generic}.

Then cl is a closure operator on M̄. In particular, it is idempotent.

Proof. Clearly cl has finite character, and A ⊆ B implies A ⊆ cl(A) ⊆ cl(B). Suppose a ∈ cl(cl(A)). Then there is a non-generic definable set ϕ(M̄, b̄, A) containing a, and for every b_{i} ∈ b̄ a non-generic definable set ϕ_{i}(M̄, A) containing b_{i}. If a were generic over A, then A would satisfy

d_{p}x∃ȳ{ϕ(x, ȳ, X) ∧ ¬d_{p}zϕ(z, ȳ, X) ∧ ⋀_{i}[ϕ_{i}(y_{i}, X) ∧ ¬d_{p}zϕ_{i}(z, X)]}.

Hence, there is A_{0} ⊆ M satisfying this formula. But in M a non-generic formula defines a finite set, so the formula

∃ȳ{ϕ(x, ȳ, A_{0}) ∧ ¬d_{p}zϕ(z, ȳ, A_{0}) ∧ ⋀_{i}[ϕ_{i}(y_{i}, A_{0}) ∧ ¬d_{p}zϕ_{i}(z, A_{0})]},

defines a finite set, which cannot be generic, a contradiction. Thus cl(cl(A)) = cl(A). □

We shall now prove a version of the dichotomy theorem for minimal structures from [9] in our context. The proof uses arguments from [5], where a similar result was proved for locally strongly regular types.

Proposition 2.2. Let (a, b) be a generic Morley sequence over M. Then exactly one of the following two cases holds.

• Symmetric: tp(a, b/M) = tp(b, a/M). In this case, a generic Morley sequence over any C ⊆ M̄ containing A is totally indiscernible over C (i.e. indiscernible over C as a set).

• Asymmetric: tp(a, b/M) ≠ tp(b, a/M). In this case, there is an M-definable strict partial order on M such that M < a < b. Moreover, (M, ≤) is a directed, well partial order having infinite increasing chains and no such chain of order type ω + 1.

Proof. Suppose that tp(a, b/M) = tp(b, a/M). This implies that for any ϕ(x, y, \overline{z}) without parameters and for any \overline{m} ∈ M we have

⊧ d_{p}x d_{p}y(ϕ(x, y, \overline{m}) ↔ ϕ(y, x, \overline{m})).

Then the same formula is satisfied by any \overline{c} ∈ C in place of \overline{m}, and so for every generic Morley sequence (a′, b′) over C we have tp(a′, b′/C) = tp(b′, a′/C). By induction, it follows easily that generic Morley sequences of any length are totally indiscernible.

Now, suppose tp(a, b/M) ≠ tp(b, a/M). We will find ϕ(x, y) ∈ tp(a, b/M) such that:

(1) ⊧ ∀x, y(ϕ(x, y) → ¬ϕ(y, x)) (ϕ(x, y) is asymmetric);
(2) ϕ(M, b) = M (i.e. ϕ(c, y) ∈ p(y) for all c ∈ M);
(3) ϕ(a, y) ⊢ p(y).

Since p is definable, it has a unique heir and a unique coheir in S_{1}(Ma); since tp(a, b/M) ≠ tp(b, a/M), the two must be distinct. Therefore tp(b/Ma) is not a coheir, so there is ϕ(a, y) ∈ tp(b/Ma) which is satisfied by no element of M. Since tp(a/Mb) is a coheir, ϕ(M, b) is infinite. Definability of p implies that ϕ(M, b) is cofinite; modifying it slightly on M, we may assume ϕ(M, b) = M. Moreover, ϕ(b, x) ∉ tp(a/Mb) as tp(a/Mb) is a coheir, so (a, b) satisfies ϕ′(x, y) = ϕ(x, y) ∧ ¬ϕ(y, x). Clearly ϕ′ satisfies conditions (1)–(3).

We leave to the reader to verify that the formula

ϕ(x, y) ∧ ∀t(ϕ(y, t) → ϕ(x, t))

defines a strict partial order; denote it by x < y. Now, we prove that for any c ∈ M we have c < b, i.e.

⊧ ϕ(c, b) ∧ ∀t(ϕ(b, t) → ϕ(c, t)).

Condition (2) implies ⊧ ϕ(c, b). Let d be such that ⊧ ϕ(b, d). Then, by (3), d realizes p, so ϕ(c, y) ∈ p(y) implies ⊧ ϕ(c, d). Thus, c < b. Since this holds for all c ∈ M and tp(a/bM) is finitely satisfiable in M, we conclude that a < b. As tp(a/M) = tp(b/M) and M < b, we also get M < a.

Now, the formula x < b belonging to tp(a/bM) is satisfied in M, say by c_{1} ∈ M. Then c_{1} < b; in fact, since M < a, we also have c_{1} < a. Hence, the formula c_{1} < x < b belongs to tp(a/bM), and so there is c_{2} ∈ M such that c_{1} < c_{2} < b; then also c_{2} < a, so the formula c_{1} < c_{2} < x < b belongs to tp(a/bM). Continuing in this way, we get an infinite increasing chain in M. We leave to the reader to verify that ≤ is a directed, well partial order with no chains of order type ω +1. □

The above proposition leads to the following definition.

Definition 1. M is symmetric if tp(a, b/M) = tp(b, a/M) for each/some generic Morley sequence (a, b) over M; equivalently, if generic Morley sequences (of arbitrary length) over any set C are totally indiscernible over C. Otherwise M is asymmetric.

From now on, whenever a definable partial order < is clear from the context, x ∼ y will be defined as ¬(x < y ∨ y < x).

Definition 2. A definable partial order < on M with infinite chains is almost linear if ∼ is an equivalence relation on p(M̄). We call M almost linear if such an order exists.

Remark 2.1. Suppose that < is a definable partial order with infinite chains on some minimal structure. Then the following conditions are equivalent.

(1) < is almost linear.

(2) After a modification of < on a finite set, incomparability ∼ becomes an equivalence relation (we allow here modifications of < between elements of this finite set and all other elements, but in such a way that the resulting order is definable).

(3) After a modification of < on a finite set, incomparability ∼ becomes an equivalence relation, and the set of equivalence classes has order type (ω, <), (ω^{∗}, <) or (ω + ω^{∗}, <).

Proof. The implications (3) → (2) → (1) are clear. In order to see that (1) → (2), one should use compactness and the minimality of M. Finally, for (2) → (3) it suffices to notice that the order type of a minimal linear order is either one of the types listed in (iii), or (ω + n, <) or ((ω + n)^{∗}, <). □

Remark 2.2. Assume that M is ordered by < with an infinite increasing chain. Then C := {c ∈ M | c < x ∈ p(x)} is a definable, co-finite subset of M, and hence < is a well partial order and M is asymmetric. After modifying < so that the elements of M \ C are below all the others, we get that C = M, and then < is a directed, well partial order having infinite increasing chains and no such chain of order type ω + 1. If in addition < is almost linear, then after modifying < on a finite set we get that M/∼ is ordered of order type ω.

Of course, analogous observations are true when M contains an infinite decreasing chain. Proposition 2.2 together with Remark 2.2 yield the following observation.

Remark 2.3. Let M be a minimal structure whose generic type is definable. Then M is asymmetric iff M is ordered.

We finish this section with an example of an ordered minimal structure which is due to Grzegorz Jagiella. Although the structure is almost linear, it has a definable order with infinite chains which is not almost linear.

Example 1. Let M = ω × {l, r}. Define an order < on M by putting the natural orders on ω × {l} and on ω × {r}, together with

(x, l) < (y, r) ⟺ x + 2 ≤ y and (y, r) < (x, l) ⟺ y + 2 ≤ x

for all natural numbers x and y.

We leave to the reader to verify that (M, <) is minimal. For n ∈ ω define a_{2n} = (2n, l) and a_{2n+1} = (2n + 1, r). Then a_{n} and a_{n+1} are incomparable for all n, and so < is not almost linear.

Now, we show that M interprets (ω, <). First, note that ‘y is maximal incomparable to x’ is a definable function f(x) = y. Then for x, y ∈ M define:

x <′ y if and only if x < y or y = f(x).

Then <′-incomparability is an equivalence relation with 2-element classes {(n, r), (n, l)}, and, after factoring it out, we end up with (ω, <). By Remark 2.1, <′ is almost linear in M, so M is almost linear.

3. The symmetric case

In this section, we shall prove that symmetric minimal fields are algebraically closed. The following is a version of the corresponding result for locally strongly regular types from [5]. It is adapted to the context of minimal structures.

Proposition 3.1. Let M be a symmetric minimal structure with definable (over ∅) generic type, M̄ its monster model and cl the closure operator from Proposition 2.1. Then (M̄, cl) is an infinite dimensional pre-geometry, and a_{1}, …, a_{n} is clindependent over A if and only if it is a generic Morley sequence over A. In particular, if (a_{1}, …, a_{n}) is a generic Morley sequence over A and a_{1}, …, a_{n} ∈ cl(A, b_{1}, …, b_{n}), then (b_{1}, …, b_{n}) is also a generic Morley sequence over A.

Proof. We only have to prove the exchange property. So consider a ∈ cl(Ab)\cl(A). Then a ∉ cl(A), so tp(a/A) is generic. Suppose b ∉ cl(Aa). Then tp(b/Aa) is generic, and (a, b) is a generic Morley sequence over A. By symmetry, (b, a) is a generic Morley sequence over A. In particular, a ∉ cl(Ab), a contradiction. Thus, cl satisfies the exchange property. □

It follows that the generic type p is generically stable, and orthogonal to all nongeneric types (see [5] for the definitions; we will not use this terminology in this paper).

Now, it is straightforward to deduce Theorem 1 from [2, Theorem 1.13], where it is proved that any field carrying a pre-geometry with certain homogeneity properties is algebraically closed; it is based on Macintyre’s proof that ω_{1}-categorical fields are algebraically closed [3]. Here, we will give an alternative proof based on an argument of Wheeler [11, Theorem 2.1]; it was also used by Pillay [4, Proposition 5.2] to prove that ω-stable fields are algebraically closed.

Lemma 3.1. If K is a minimal field, then (K^{∗})^{n} = K for any n > 0. In particular K is perfect.

Proof. (K^{∗})^{n} is an infinite subgroup of K^{∗}, so, by minimality, it is co-finite and (K^{∗})^{n} = K^{∗}. □

Theorem 1. A symmetric minimal field is algebraically closed.

Proof. Let K be a symmetric minimal field, K̄ ≻ K a monster model, and p the generic type. Let F denote an algebraic closure of K̄. Suppose for a contradiction that some α_{1} ∈ F \K is algebraic over K. Let f(x) = x^{n}+a_{n−1}x^{n−1}+⋯+a_{0} ∈ K[x] be the minimal polynomial of α_{1} over K. Since K is perfect, f has n pairwise distinct roots α_{1}, …, α_{n} in F. Let (t_{0}, …, t_{n−1}) ∈ K̄^{n} be a Morley sequence in p over K, and define r_{i} = t_{0} + t_{1}α_{i} + ⋯ + t_{n−1}α_{i}^{n−1} for i = 1, …, n. Then

( \begin{matrix}1 & α_{1} & … & α_{1}^{n−1} \\ 1 & α_{2} & … & α_{2}^{n−1} \\ ⋮ & ⋮ & … & ⋮ \\ 1 & α_{n} & … & α_{n}^{n−1} \end{matrix} ) ( t_{0} ; t_{1} ; ⋮ ; t_{n−1} ) = ( r_{1} ; r_{2} ; ⋮ ; r_{n} )

and, since the matrix is invertible, (r_{1}, …, r_{n}) and (t_{0}, …, t_{n−1}) are interalgebraic over K.

Let c_{0}, …, c_{n−1} be the symmetric functions of r_{1}, …, r_{n}. Then the sequences (c_{0}, …, c_{n−1}) and (r_{1}, …, r_{n}) are interalgebraic over K, hence (t_{0}, …, t_{n−1}) and (c_{0}, …, c_{n−1}) are interalgebraic over K, too. Since c_{i}, t_{j} are in K̄, we can apply Proposition 3.1 to conclude that (c_{0}, …, c_{n−1}) is a Morley sequence in p over K. But for generic x over c_{1}, …, c_{n−1} the element

c′ = −(x^{n} + c_{n−1}x^{n−1} + ⋯ + c_{1}x)

is again generic by Proposition 3.1, so it has the same type over c_{1}, …, c_{n−1} as c_{0}, and there is an automorphism σ fixing c_{1}, …, c_{n−1} and moving c′ to c_{0}. Then σ(x) is a zero of the polynomial

z^{n} + c_{n−1}z^{n−1} + ⋯ + c_{1}z + c_{0}.

Thus there is i such that

σ(x) = r_{i} = t_{0} + t_{1}α_{i} + ⋯ + t_{n−1}α_{i}^{n−1} ∈ K̄.

This means that the degree of the minimal polynomial of α_{i} over K(t_{0}, …, t_{n−1}) is smaller than n. Since K(t_{0}, …, t_{n−1}) ⊆ K̄, this implies that the degree of the minimal polynomial of α_{i} over K̄ is also smaller than n. On the other hand, as K ≺ K̄ and f(x) ∈ K[x] is the minimal polynomial of α_{i} over K (so it is irreducible in K[x]), we get that f(x) is also irreducible in K̄[x], and so it is the minimal polynomial of α_{i} over K̄. This is a contradiction, because deg(f) = n. □

Notice the following consequence of Proposition 2.2 and Theorem 1.

Corollary 3.1. Each minimal field whose theory does not have the strict order property is algebraically closed. In particular, each minimal field whose theory is simple is algebraically closed.

4. Asymmetric minimal groups

In this section, we shall show some general properties of asymmetric minimal groups. In particular, an almost linear minimal group is either elementary abelian or a finite sum of Prüfer p-groups for some prime p; it follows that there is no almost linear minimal field. Unfortunately, as far as asymmetric minimal groups in general are concerned, the following questions are still open; an affirmative answer would immediately imply Podewski’s conjecture:

Question 3. Is every asymmetric minimal group almost linear? Is it at least torsion?

Given a minimal group G, a generic element is an element which is generic over G, i.e. a realization (in a monster model Ḡ) of the unique generic type p ∈ S_{1}(G). By Proposition 2.2 and Remark 2.2, whenever we are working in an asymmetric minimal group (G, <, +, 0, …), we can and do assume that < is a directed, well, strict partial order with an infinite increasing chain and with no such chain of order type ω + 1, and such that G < g for any generic g.

Lemma 4.1. Let (G, <, +, 0, …) be an asymmetric minimal group, g generic, g_{1}, …, g_{k} < g, and n_{1}, …, n_{k} integers. Then n_{1}g_{1} + ⋯ + n_{k}g_{k} ≠ g.

Proof. Suppose n_{1}g_{1} + ⋯ + n_{k}g_{k} = g. Then

(4.1)  ∃x_{1}, …, x_{k} (⋀_{i=1}^{k} x_{i} < x ∧ ∑_{i=1}^{k} n_{i}x_{i} = x)

holds generically, and hence outside a finite set X ⊂ G. By Reineke’s result, G is either elementary abelian, or abelian divisible. In either case, any finitely generated subgroup is a proper subgroup of G. Let H be the subgroup generated by X. Consider a minimal element a in G \ H. As (4.1) holds outside X ⊆ H, it is satisfied by a. But any a_{1}, …, a_{k} < a are in H by the minimality of a, as is ∑_{i=1}^{k} n_{i}a_{i}. So ∑_{i=1}^{k} n_{i}a_{i} ≠ a, a contradiction. □

There is also a corresponding version for asymmetric minimal fields:

Lemma 4.2. Let (K, <, +, ·, 0, 1, …) be an asymmetric minimal field, g generic, and g_{1}, …, g_{k} < g. Let f(x_{1}, …, x_{k}) be a rational function over K such that the tuple (g_{1}, …, g_{k}) is in its domain. Then f(g_{1}, …, g_{k}) ≠ g.

Proof. Suppose f(g_{1}, …, g_{k}) = g. Then

(4.2)  ∃\overline{x} = (x_{1}, …, x_{k}) (⋀_{i=1}^{k} x_{i} < x ∧ \overline{x} ∈ dom(f) ∧ f(\overline{x}) = x)

holds generically, and hence outside a finite set X ⊂ K. Let F be the subfield of K generated by X. Since K is minimal, it is closed under n-th roots for all n > 0 by Lemma 3.1 and thus not finitely generated. Hence F is a proper subfield; consider a minimal element a in K \ F. As (4.2) holds outside X ⊆ F, it is satisfied by a. Consider any a_{1}, …, a_{k} < a such that (a_{1}, …, a_{k}) ∈ dom(f). Then a_{1}, …, a_{k} ∈ F by the minimality of a, hence f(a_{1}, …, a_{k}) ∈ F, and so f(a_{1}, …, a_{k}) ≠ a, a contradiction. □

Recall that a divisible abelian group G splits as a direct sum of the torsion subgroup T or(G) and a direct sum of copies of ℚ; furthermore, T or(G) is a direct sum of some numbers of copies of Prüfer p-groups, where p ranges over prime numbers. Under the assumption that G is not a finite sum of Prüfer groups, we can strengthen the conclusion of Lemma 4.1.

Lemma 4.3. Let (G, <, +, 0, …) be an asymmetric minimal group. Let g be generic, g_{1}, …, g_{k} < g, and n_{1}, …, n_{k}, n integers with ng ≠ 0. Then n_{1}g_{1} + ⋯ + n_{k}g_{k} ≠ ng, or G is a finite sum of Prüfer p-groups for some primes p dividing n (possibly with repetitions).

Proof. The proof is very similar to the previous one. Assuming for a contradiction that n_{1}g_{1} + ⋯ + n_{k}g_{k} = ng, we get that the set X ⊂ G of realizations of the negation of the formula

(4.3)  ∃x_{1}, …, x_{k} (⋀_{i=1}^{k} x_{i} < x ∧ ∑_{i=1}^{k} n_{i}x_{i} = nx)

is finite. Let H be the n-divisible hull of the subgroup generated by X, i.e., the collection of all elements h ∈ G such that n^{m}h ∈ ⟨X⟩ for some m ∈ ℕ. If H ≠ G, we finish as before by considering a minimal element a ∈ G \ H. So it remains to show that H is a proper subgroup of G. By Reineke’s result, either G is elementary abelian of prime exponent p and H is finite (note that ng ≠ 0 implies that p does not divide n), or G is divisible with only finitely many elements of any given finite order. If it contains a copy Q of ℚ, then (n + 1)^{−k}1_{Q} ∉ H for sufficienly big k. Otherwise G contains a copy P of a Prüfer p-group for some p not dividing n, so H ∩ P must be finite. □

Corollary 4.1. Let (G, <, +, 0, …) be an asymmetric minimal group. If g is generic and h < g, then ±g ± h ∼ g.

Proof. Fix any choice of ± in ±g ± h. If ±g ± h > g, we have that g, h < ±g ± h and ±g ± h is generic; if ±g ± h < g, we have h, ±g ± h < g; in either case, we contradict Lemma 4.1. □

Corollary 4.2. Let (G, <, +, 0, …) be an asymmetric minimal group and let g be generic. Then g ≮ ng for any integer n. If ng ≠ 0 and G is not a sum of finitely many Prüfer p-groups for primes p dividing n, then ng ≮ g, whence g ∼ ng. Moreover, g ∼ g′ for any g′ ∈ \frac{1}{n}g.

Proof. If ng > g, then clearly ng is generic and it is a sum of strictly smaller elements, which contradicts Lemma 4.1. If ng < g, we take g_{1} = ng and n_{1} = 1, contradicting Lemma 4.3. Thus g ∼ ng. If g′ ∈ \frac{1}{n}g, then g′ is generic, so g′ ∼ ng′ = g by the first part of the proof. □

Lemma 4.4. Let G be an asymmetric minimal group, not of exponent dividing n, and g generic. Then all elements of \frac{1}{n}g are ∼-related.

Proof. The finite set X := {g′ : ng′ = g} must have a minimal element g_{0} and a maximal element g_{1}, both generic over G. But

X = {g′ : ng′ = ng_{0}} = {g′ : ng′ = ng_{1}}

is invariant under an automorphism taking g_{0} to g_{1}. Hence g_{0} ∼ g_{1}. □

Theorem 2. An almost linear minimal group G is either elementary abelian, or it is a sum of finitely many Prüfer p-groups for a fixed prime p. In particular, G is a torsion group.

Proof. Suppose G is a counter-example. Then G divisible, and at least one of the following cases holds:

1. G contains a copy P of a Prüfer p-group, but is not a sum of Prüfer pgroups,

2. G contains a copy P of ℚ; in this case, put p = 2. In any of these cases, by Corollary 4.2, the set

X := \{x ∈ G : x ≁ y for some y ∈ \frac{1}{p}x\}

is finite; let H be the subgroup generated by X. Then H ∩ P is a proper subgroup of P, so there is a_{0} ∈ P \ H. Choose a_{k} ∈ P with pa_{k+1} = a_{k} for all k < ω. Then all a_{k}’s are outside H, so transitivity of ∼ implies that {a_{k} : k < ω} is a ∼-antichain. Since it must be finite, there is j > i with a_{i} = a_{j}. Then p^{j−i}a_{0} = a_{0}, a contradiction with the fact that the order of a_{0} is a power of p in Case 1 and is infinite in Case 2. □

This implies immediately the non-existence of almost linear minimal fields.

Theorem 3. There is no almost linear minimal field.

Proof. The multiplicative group of an infinite field is neither elementary abelian, nor a finite sum of Prüfer p-groups for a fixed prime p. □

5. Almost linear minimal groups as valued groups

Recall that a valued abelian group is an abelian group G together with a surjective valuation v : G → Γ, where Γ is a linearly ordered set with maximum ∞, such that:

(1) v(x) = ∞ if and only if x = 0.

(2) v(x − y) ≥ min{v(x), v(y)}. Note that the axioms imply v(−x) = v(x) and v(x − y) = min{v(x), v(y)} unless v(x) = v(y). It follows that for every γ ∈ Γ \ {∞} the sets B̄(γ) = {x ∈ G : v(x) ≥ γ} and B^{o}(γ) = {x ∈ G : v(x) > γ} are subgroups of G. Valued abelian groups have been studied by Simonetta [8] and de Aldama [1], who consider the following conditions for all primes p:

(3) ∀x, y [v(px) < v(py) → v(x) < v(y)].

(4) ∀x, y [v(x) < v(y) → (v(px) < v(py) ∨ px = 0)].
(5) ∀x, y [v(x) < v(py) ∨ ∃z pz = x].

As v is surjective, for all n ∈ ℕ^{∗} one can define a function f_{n} : Γ → Γ as f_{n}((v(x)) = v(nx). In addition, for every m ∈ ℕ^{∗} we consider the unary relation R_{m} on Γ given by

R_{m}(x) ⇔ |B̄(x)/B^{o}(x)| > m.

It follows from the axioms (1)–(5) that f_{n} is well-defined and increasing, strictly so on Γ \ f_{n}^{−1}(∞). Moreover, if f_{n}(γ) ≠ ∞, then R_{m}(γ) ⇔ R_{m}(f_{n}(γ)). We put

L_{vg} = {+, 0, v, ≤, ∞} and L_{v} = {≤, R_{n}, f_{n} : n ∈ ℕ^{∗}}.

Simonetta shows that if G is a valued abelian group satisfying (1)–(5), then there is at most one prime p such that G is not p-divisible, and at most one prime q such that G has q-torsion. Moreover, he obtains the following relative quantifier elimination result:

Fact 5.1. [8, Theorem 3.3] Every L_{vg}-formula ϕ(x̄, ȳ) with variables x̄ in the group sort and variables ȳ in the value sort is equivalent in G to some formula ϕ_{v}(v(t_{1}(x̄)), …, v(t_{n}(x̄)), ȳ), where the t_{i}(x̄) are group terms in x̄ and ϕ_{v} is an L_{v}-formula. Moreover, ϕ_{v} and t_{1}, …, t_{n} only depend on p and q.

Clearly a valued group with infinite value set is almost linear, where we take the inverse order induced from the valuation. For minimal groups, we shall now prove the converse: An almost linear minimal group G carries interdefinably the structure of a valued group. Recall that by Remark 2.2 we can and do choose a definable order on G so that G/∼ is ordered in type ω.

Lemma 5.1. Let G be an almost linear minimal group and put H_{g} = {x ∈ G : x ≯ g}. Then H_{g} is a subgroup for almost all g ∈ G. The collection of these subgroups is linearly ordered by inclusion in order type ω.

Proof. Let g be generic and x, y ≯ g. Suppose for a contradiction that x ± y > g. By transitivity of ∼, we have x, y < x ± y, contradicting Lemma 4.1. Therefore H_{g} is a subgroup for generic g, and thus for almost all g by minimality. Clearly g < g′ implies H_{g} ⊂ H_{g′}, and g ∼ g′ implies H_{g} = H_{g′} by transitivity of ∼. Thus the set {H_{g} : g ∈ G} has the same order type with respect to inclusion as {g/∼ : g ∈ G} with respect to <, namely ω. □

Proposition 5.1. Let G be an almost linear minimal group. After modifying the order on a finite set, the map v : G ↦ G/∼ endows G with the structure of a valued abelian group, where G/∼ is ordered by g/∼ > g′/∼ if g < g′, of order type ω^{∗}. In particular, G as almost linear group and G as valued group are interdefinable (with parameters).

Proof. Let X ⊆ G be the co-finite set of g ∈ G such that H_{g} is a subgroup. We modify the order on G so that 0 is the unique minimal element. This can only increase X; in particular, 0 ∈ X and H_{0} = {0}. Let g_{0} be a minimal element greater than G \ X. We modify the order by making all elements of the finite set H_{g_{0}} \ {0} incomparable. These modifications are clearly definable, the modified order on G is still almost linear, and H_{g} is now a subgroup of G for all g ∈ G. Of course, g < g′ still implies H_{g} ⊂ H_{g′}, and g ∼ g′ yields H_{g} = H_{g′}.

Now, H_{0} = {0} implies axiom (1); the fact that H_{g} is a subgroup for all g ∈ G yields axiom (2). As we have only modified an initial segment of the order on G, the set of equivalence classes G/∼ still has order type ω^{∗}. We have obtained the valuation v definably from the almost linear structure; inversely, as we only modified the order on finitely many elements, we can define the original almost linear structure from the valuation. □

We say that Axioms (3)-(5) hold generically if they hold outside a finite set. It is clear that if Axiom (3) holds outside a finite set, then each function f_{n} is well-defined outside a finite set D_{n}; we extend f_{n} to the whole of Γ by putting f_{n}(D_{n}) = ∞.

Let G be a minimal valued group. Since B^{o}(v(g)) is a proper definable subgroup of G and thus finite for all g ∈ G, it follows that Γ has order type ω^{∗} or is finite. In the latter case, if γ ∈ Γ is the minimal element, then B̄(γ) = G, so the valuation is determined by the restriction of v to the finite group B^{o}(γ) and hence definable in the pure group structure. Henceforth, we shall assume that Γ has order type ω^{∗}, so G is almost linear; by Theorem 2 it is either elementary abelian of exponent p or a finite product of Prüfer p-groups, for some prime p. Note that for n coprime to p the function f_{n} is just the identity as all groups B̄ and B^{o} are finite p-groups; this also shows that v has finite fibres. Clearly, if G has exponent p, then f_{p} maps Γ to ∞. We write g ∼ g′ if v(g) = v(g′).

Lemma 5.2. If G is divisible and g generic, then B̄(v(g))/B̄(v(pg)) is finite and isomorphic to B̄(v(pg))/B̄(v(p^{2}g)) as valued groups (with the induced valuation), via the map induced by x ↦ px. In particular, the interval [v(g), v(pg)] is finite and L_{v}-isomorphic to [v(pg), v(p^{2}g)]. Hence for all but finitely many γ the function f_{p} is well-defined and corresponds to a right shift by ℓ, where [v(g), v(pg)] has length ℓ + 1. It is thus definable from the order.

Proof. Let us check first that if g is generic and g ∼ g′, then pg ∼ pg′. So consider X := {pg′ : g′ ∼ g}; it must have a minimal element g_{0} and a maximal element g_{1}, both generic over G. By Lemma 4.4, all elements of \frac{1}{p}g_{0} and \frac{1}{p}g_{1} are ∼-related to g, so

X = \{pg′ : g′ ∼ \frac{1}{p}g_{0}\} = \{pg′ : g′ ∼ \frac{1}{p}g_{1}\}

is invariant under an automorphism taking g_{0} to g_{1}. Hence g_{0} ∼ g_{1}.

Next, let us check that pg ∼ pg′ implies g ∼ g′ for generic g. So consider a minimal element g_{0} and a maximal element g_{1} of the set X := {g′ : pg′ ∼ pg}. Since

X = {g′ : pg′ ∼ pg_{0}} = {g′ : pg′ ∼ pg_{1}}

and the generic type is unique, we have g_{0} ∼ g_{1}.

It follows that x ↦ px maps B̄(v(g)) \ B^{o}(v(g)) onto B̄(v(pg)) \ B^{o}(v(pg)). As it is a group homomorphism, it also maps B̄(v(g)) onto B̄(v(pg)), and hence B^{o}(v(g)) onto B^{o}(v(pg)). Thus g′ < g if and only if pg′ < pg for all g′. We conclude that the map induced by x ↦ px is an isomorphism from B̄(v(g))/B̄(v(pg)) to B̄(v(pg))/B̄(v(p^{2}g)) as valued groups, and that f_{p} yields an L_{v}-isomorphism between [v(g), v(pg)] and [v(pg), v(p^{2}g)]. By minimality, all of this holds for all g outside some finite set Y.

It remains to show that B̄(v(g))/B̄(v(pg)) is finite. Let g_{0} ∈ G be such that G[p] ∪ Y ⊆ B̄(v(g_{0})) (where G[p] := {x ∈ G : px = 0}). Then for all h > g_{0} the map x ↦ px maps the finite group B̄(v(h)) onto the subgroup B̄(v(ph)) and has finite kernel G[p] independent of h. So |B̄(v(h))/B̄(v(ph))| = |G[p]| for all h > g_{0}. □

An inspection of Simonetta’s proof of Fact 5.1 shows that it yields the following proposition, as finite sets of exceptions can be dealt with definably.

Proposition 5.2. Let G be a valued abelian group whose valuation v has finite fibres and whose value set Γ has order type ω^{∗}. Assume that G is either elementary abelian of exponent p or a finite product of Prüfer p-groups for some prime p, and Axioms (3) and (4) hold generically, i.e. outside a finite set. Then every L_{vg}- formula ϕ(x̄, ȳ) with variables x̄ in the group sort and variables ȳ in the value sort is equivalent to a formula ϕ_{v}(v(t_{1}(x̄)), …, v(t_{n}(x̄)), ȳ), where the t_{i}(x̄) are terms in x̄ and ϕ_{v} is an L_{v}-formula (both with parameters).

Notice that by Lemma 5.2, the assumptions of Proposition 5.2 are satisfied for any valued minimal group with infinite value set Γ.

Theorem 4. A valued abelian group G with infinite value set Γ is minimal if and only if the induced L_{v}-theory on Γ is minimal of order type ω^{∗}, the map v has finite fibres, and either G is elementary abelian, or a finite product of Prüfer p-groups for some prime p and f_{p} is eventually a well-defined L_{v}-isomorphism acting by right shift.

Proof. If G is minimal as a valued group, then Γ with the induced structure must be minimal, since an infinite co-infinite subset X of Γ has an infinite co-infinite preimage v^{−1}(X). We have seen above that Γ has order type ω^{∗}, the group is either elementary abelian or a finite product of Prüfer p-groups, and v has finite fibres. The fact that f_{p} is eventually an L_{v}-isomorphism acting by right shift follows from Lemma 5.2.

Conversely, suppose that the L_{v}-structure Γ is minimal of order type ω^{∗}, all fibres of v are finite, and G is elementary abelian of exponent p, or a finite product of Prüfer p-groups and f_{p} is eventually an L_{v}-isomorphism acting by right shift.

This implies that f_{n} is the identity for n coprime to p, and f_{p}(Γ) = ∞ if G has exponent p. All of this implies that Axioms (3)-(5) hold generically.

Consider a formula ϕ(x, ḡ), where ḡ are parameters in G. (Clearly, we can replace any parameter γ ∈ Γ by some element of v^{−1}(γ).) By Proposition 5.2 (enlarging the tuple ḡ of parameters if necessary), this formula is equivalent to a formula ϕ_{v}(v(t_{1}(x, ḡ)), …, v(t_{k}(x, ḡ))), where ϕ_{v} is an L_{v}-formula and t_{1}, …, t_{k} are group terms. As group terms are just ℤ-linear combinations, there are integers n_{i} ∈ ℤ and h_{i} ∈ ⟨g_{1}, …, g_{k}⟩ < G such that t_{i}(x, ḡ) = n_{i}x + h_{i}; if G has exponent p, we may choose 0 ≤ n_{i} < p. Since v has finite fibres, Γ has order type ω^{∗} and G is elementary abelian or has finite n-torsion for all n, the set

X = {g ∈ G : v(n_{i}g) ≥ v(h_{i}) for some i with n_{i} ≠ 0}

is finite. Let Y be a finite subset of G such that all the f_{n_{i}} for n_{i} > 0 are welldefined outside v(Y). On G \ (X ∪ Y) the formula ϕ_{v}(v(t_{1}(x, ḡ)), …, v(t_{k}(x, ḡ))) is equivalent to

ϕ_{v}(f_{n_{1}}(v(x)), …, f_{n_{k}}(v(x))) = ϕ′_{v}(v(x), γ̄),

where we have put f_{n_{i}}(v(x)) = v(h_{i}) = γ_{i} ∈ Γ whenever n_{i} = 0. Since ϕ′_{v}(y, γ̄) defines a finite or co-finite subset of Γ and the fibres of v are finite, ϕ′_{v}(v(x), γ̄) defines a finite or co-finite subset of G. It follows that ϕ(x, ḡ) defines a finite or co-finite subset of G. Thus G is minimal as a valued group. □

This yields a classification of valued minimal groups:

Theorem 5. A valued group G with infinite value set Γ is minimal if and only if (1) Γ has order type ω^{∗},

(2) v has finite fibres,

(3) either G is elementary abelian of exponent p and (a) either there is n_{0} < ω such that R_{p^{n_{0}}} ∧ ¬R_{p^{n_{0}+1}} is co-finite,

(b) or R_{p^{n}} ∧ ¬R_{p^{n+1}} is finite for all n < ω,

(4) or G is a finite product of Prüfer p-groups, f_{p} is eventually an L_{v}-isomor-phism acting by right shift, and there is n_{0} < ω such that R_{p^{n_{0}}} ∧ ¬R_{p^{n_{0}+1}} almost everywhere.

Proof. Clearly the conditions are necessary for Γ to be minimal as an L_{v}-structure; it is easy to see that they are also sufficient. Note that the analogue of option (3)(b) cannot occur in case (4), as |B̄(v(g))/B̄(v(pg))| remains bounded by Lemma 5.2. □

Note that all cases can be easily realised as a valued abelian group, which hence must be almost linear minimal. In particular, we obtain examples of almost linear minimal groups which are elementary abelian of exponent p (for any prime p) as well as examples of almost linear minimal torsion groups of infinite exponent, namely finite products of Prüfer p-groups (for any fixed prime p). This shows that the conclusion of Theorem 2 is strongest possible.

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[8] P. Simonetta, Abelian C-minimal groups, Ann. Pure Applied Logic, vol. 110/1–3 (2001), 1–22

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Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

   E-mail address: kkrup@math.uni.wroc.pl

Mathematical Institute SANU, Knez Mihajlova 36, Belgrade, Serbia

   E-mail address: tane@mi.sanu.ac.rs

Université de Lyon; CNRS; Université Lyon 1; Institut Camille Jordan UMR5208, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France

   E-mail address: wagner@math.univ-lyon1.fr