Permutation groups with small orbit growth

Manuel Bodirsky; Bertalan Bodor

doi:10.1515/jgth-2018-0220

Publicly Available Published by De Gruyter January 20, 2021

Permutation groups with small orbit growth

Manuel Bodirsky and Bertalan Bodor

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2018-0220

Abstract

Let 𝒦exp+ be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c⁢nd⁢n orbits in its componentwise action on the set of n-tuples with pairwise distinct entries, for some constants c,d with d<1. We show that 𝒦exp+ is precisely the class of finite covers of first-order reducts of unary structures, and also that 𝒦exp+ is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from 𝒦exp+. We also show that Thomas’ conjecture holds for 𝒦exp+: all structures in 𝒦exp+ have finitely many first-order reducts up to first-order interdefinability.

1 Introduction

A first-order reduct of a structure 𝔄 is a relational structure with the same domain as 𝔄 whose relations are first-order definable over 𝔄. Simon Thomas conjectured that every homogeneous structure 𝔄 with finite relational signature has only finitely many first-order reducts up to first-order interdefinability [64]. The conjecture has been verified for many famous homogeneous structures 𝔄: e.g., for the ordered rationals [28], the countably infinite random graph [64], the homogeneous universal Kn-free graphs [65], the expansion of (ℚ;<) by a constant [50], the universal homogeneous partial order [59], and the random ordered graph [20], and many more [1, 2, 13, 11]. If we drop the assumption that the signature of the homogeneous structure 𝔄 is relational, then the conjecture of Thomas is false even if we keep the assumption that 𝔄 is ω-categorical: already the countable atomless Boolean algebra has infinitely many first-order reducts [23].

Thomas’ conjecture highlights our limited understanding of the class of homogeneous structures 𝔄 with finite relational signature. One approach to widen our understanding is to study homogeneous structures for some fixed signature; for example, classifications exist for the class of all homogeneous tournaments [52], homogeneous undirected graphs [44], homogeneous partial orders [63], general homogeneous digraphs [35], homogeneous permutations [32], and homogeneous coloured multipartite graphs [49, 53]. However, already the class of homogeneous 3-uniform hypergraphs appears to be very difficult [5]. If we impose additional assumptions, e.g., that the age of 𝔄 can be described by finitely many forbidding substructures, we might hope for systematic understanding and effectiveness results for various questions. However, it is not clear how to use this assumption for proving that 𝔄 has finitely many first-order reducts.

Another approach to understanding the class of homogeneous structures, followed in this paper, is to start with the most symmetric structures in this class. Symmetry can be measured by the number of orbits oni⁢(G) of the diagonal action of the automorphism group G=Aut⁡(𝔄) on tuples from An that have pairwise distinct entries. By the theorem of Engeler, Ryll-Nardzeski, and Svenonius, these orbits are in one-to-one correspondence with the model-theoretic types of n pairwise distinct elements in 𝔄. Alternatively, we might count the number of orbits ons⁢(G) of the action of G on n-element subsets of A. The investigation of both of these measures has been pioneered by Cameron; see [30] for an introduction to the subject. The sequence oni⁢(G) is linked to labeled enumeration problems, which are the most intensively studied counting problems in enumerative combinatorics, while ons⁢(G) is linked to unlabeled enumeration problems. Many structural results about G are available when we impose restrictions on ons⁢(G); see, e.g., [55, 54, 56]. The present article, in contrast, focuses on restricting oni⁢(G).

A structure 𝔄 is finite if and only if oni⁢(Aut⁡(𝔄)) is eventually 0. It is a by-product of our results that the class 𝒦exp of all structures 𝔄 where oni⁢(Aut⁡(𝔄)) grows at most exponentially equals the class of first-order reducts of unary structures; by a unary structure, we mean any at most countable structure with finitely many unary relations. Our main result pushes this further: we study the class 𝒦exp+ of structures 𝔄 such that oni⁢(Aut⁡(𝔄)) is bounded by c⁢nd⁢n for some constants c,d with d<1. Note that, for example, the structure (ℚ;<) does not belong to 𝒦exp+ because oni⁢(ℚ;<)=n!. Also, 𝒦exp+ contains no structure 𝔄 with a definable equivalence relation with infinitely many infinite classes because oni⁢(𝔄) would in this case be at least as large as the n-th Bell number, which grows asymptotically faster than c⁢nd⁢n (see Lemma 6.1). We show that 𝒦exp+ contains precisely those structures that are finite covers of first-order reducts of unary structures (see Theorem 6.29).

Finite covers in model theory and infinite permutation groups have been studied in the context of classifying totally categorical structures [4, 47, 48] and, more generally, for studying ω-categorical ω-stable structures [33, 34]. Finite covers became an important topic in its own [38, 40, 61]; we refer to the survey article for an introduction [39]. It follows from our result that the class of finite covers of reducts of unary structures equals the class of first-order reducts of finite covers of unary structures. Using the terminology of [39], we show that all finite covers of unary structures split, but not necessarily strongly. All structures in 𝒦exp+ can be expanded to structures that are homogeneous in a finite relational language, and we show that they all satisfy Thomas’ conjecture (see Theorem 6.37). The proof uses a result of Macpherson which implies that structures in 𝒦exp+ which have a primitive automorphism group must be highly transitive [55].

The class 𝒦exp+ can be seen as the “smallest reasonably robust class that contains all finite structures as well as some infinite ones” (for formalisations of this statement, see Section 8.3). So, whenever a statement that holds for all finite structures needs to be generalised to a class of “slightly infinite structures”, it might be a good idea to try to first prove the statement for 𝒦exp+. This is precisely the situation for the constraint satisfaction problem.

1.1 Complexity of constraint satisfaction

Let 𝔅 be a structure with finite relational signature. The constraint satisfaction problem for 𝔅 is the computational problem of deciding whether a given finite structure 𝔄 with the same signature as 𝔅 has a homomorphism to 𝔅. For finite structures 𝔅, Feder and Vardi [41] conjectured that the computational complexity of CSP⁡(𝔅) satisfies a dichotomy: it is either in P or NP-complete. Using concepts and techniques from universal algebra, Bulatov and Zhuk recently presented independent proofs of this conjecture [27, 66].

The universal-algebraic approach can also be applied when 𝔅 is countably infinite and ω-categorical. In this case, the computational complexity of 𝔅 is captured by the polymorphism clone of 𝔅 (see [16]), which can be seen as a generalisation of the automorphism group of 𝔅: it consists of all homomorphisms from 𝔅n to 𝔅, for n∈ℕ. Moreover, every ω-categorical structure 𝔅 is homomorphically equivalent to an (up to isomorphism unique) structure ℭ with the property that the automorphisms of ℭ lie dense in the endomorphisms of ℭ, called the model-complete core of 𝔅. The model-complete core ℭ of 𝔅 is again ω-categorical, and has the same CSP as 𝔅, so that we prefer to analyse ℭ rather than 𝔅. This simplification of the classification problem is a key step for many results (see, e.g., [14, 18, 6]), including the finite-domain classification [27, 66].

Therefore, if we want to classify the computational complexity of CSP⁡(𝔅) for all structures 𝔅 from a class 𝒞, it is important whether the class 𝒞 is closed under the formation of model-complete cores. When ℭ is the model-complete core of 𝔅, then it is easy to see that oni⁢(ℭ)≤oni⁢(𝔅); hence, in particular, the classes 𝒦exp and 𝒦exp+ are closed under taking model-complete cores. This makes these classes attractive goals for extending the mentioned dichotomy result from finite domains.

As mentioned before, our results imply that every structure in 𝒦exp is a first-order reduct of a unary structure. For those structures, it has already been shown that they are in P or NP-complete [15] (using the mentioned dichotomy for finite-domain CSPs). Our main result states that 𝒦exp+ is precisely the class of first-order reducts of finite covers of unary structures. For classifying the complexity of the CSP for all structures in this class, our result implies that we can assume without loss of generality that these structures are model-complete cores. We thus see our result as a first step towards classifying the CSP for first-order reducts of finite covers of unary structures.

1.2 Definable sets with atoms

In theoretical computer science, one is interested in finite representations of infinite structures; one approach to this is the framework of definable sets and computation with atoms [24, 25]. This leads to new models of computation over infinite structures with interesting links to long-standing open problems in finite model theory, namely the question whether there is a logic for P and computation in choiceless polynomial time [26].

If the “atom structure” is (ℕ;=) (which is besides (ℚ;<) the most frequently used base structure in this area), then definable sets (in this case also studied under the name nominal sets [42]) correspond precisely to the class 𝒦= of structures that are first-order interpretable over (ℕ;=) in the sense of model theory (for an explicit discussion of the connection, see [51, Lemma 7 and the remarks thereafter]). The class 𝒦= might appear to be trivial to many model theorists (all structures in it are ω-categorical, ω-stable, and they are first-order reducts of homogeneous finitely bounded structures), but in fact, many questions about this class remain open; see Section 10 for a small sample of open problems. It follows from our results (see Remark 6.30) that 𝒦exp+⊆𝒦=, and we can answer for 𝒦exp+ many questions that we cannot answer for the class 𝒦= in general. So our results can also be seen as a first step towards a better understanding of 𝒦=.

2 Preliminaries

If ∼ is an equivalence relation on X and x∈X, then [x]∼ denotes the equivalence class of x with respect to ∼, and X/∼:={[x]∼∣x∈X} denotes the set of all ∼-classes. We write |∼| for |X/∼|. If ∼1 and ∼2 are equivalence relations on X, then we say that ∼1 is finer than ∼2 (or ∼2 is coarser than ∼1) if ∼1 is contained in ∼2 (as binary relations).

2.1 Permutation group notation

When G is a group, we write H≤G if H is a subgroup of G, and H◁G if H is a normal subgroup of G. We write [G:H] for the index of H in G. For any set X, we write Sym⁡(X) for the group of all permutations of X. If G≤Sym⁡(X) and x∈X, then Gx denotes the stabiliser of the element x. Let Y⊆X. Then

GY denotes the pointwise stabiliser, and
G{Y} denotes the setwise stabiliser of the set Y.
G|Y denotes the restriction of G to Y provided that Y is preserved by G.

If Y is finite, say Y={x1,…,xn}, then we also use the notation Gx1,…,xn for the pointwise stabiliser of the set Y. Let G be a permutation group on X. An orbit of G is a set of the form {g⁢(x)∣g∈G} for some x∈X. The algebraic closure of Y⊆X with respect to G is the union of the finite orbits of GY, and it is denoted by aclG⁡(Y). If x∈X, then we use the notation aclG⁡(x) instead of aclG⁡({x}). It is well known that aclG is a closure operator on the subsets of X, and in particular, we have aclG⁡(aclG⁡(Y))=aclG⁡(Y) for all Y⊆X. If the group G is clear from the context, then we will omit the subscript from this notation.

An equivalence relation ∼ of X is called a congruence of a permutation group G⊆Sym⁡(X) if x∼y and g∈G implies g⁢(x)∼g⁢(y) for all x,y∈X and g∈G. In other words, an equivalence relation ∼ is a congruence if the corresponding partition is G-invariant. Every permutation group G⊆Sym⁡(X) has two trivial congruences, namely X2 and {(x,x)∣x∈X}. We call the former the universal congruence and the latter the identity congruence. If ∼ is a congruence of some permutation group G⊆Sym⁡(X), then G acts naturally on X/∼. The image of this action, as a subgroup of Sym(X/∼), is denoted by G/∼.

Definition 2.1.

Let π:A→B be a map. We write ∼π for the equivalence relation {(a1,a2)∣π⁢(a1)=π⁢(a2)} on A. If G is a permutation group on A such that ∼π is a congruence of G, then π gives rise to a homomorphism μπ:G→Sym⁡(B) defined by μπ(g)(a):=π(g(π-1(a))) (this is well-defined since G preserves ∼π).

2.2 Direct products

Let I be a set. For each i∈I, let Ai be a group. Then ∏i∈IAi denotes the direct product of the Ai; i.e., the elements have the form (ai)i∈I for ai∈Ai, and group composition is defined pointwise. When the Ai are permutation groups on disjoint sets Xi for every i∈I, then A:=∏i∈IAi acts naturally (intransitively) on X:=⊔i∈IXi as follows: for α∈A and x∈X, define α(x):=αi(x) if x∈Xi. It is easy to see that if each of the Ai is closed in Sym⁡(Xi), then the permutation group defined by the action of A on X is closed in Sym⁡(X), and hence is the automorphism group of some relational structure with domain X.

2.3 Wreath products

Let A be a group acting on the set F, and let Y be a set. Let H be a group acting on Y, and let X:=F×Y. Then there are natural actions of the groups N:=∏y∈YA and H on the set X, defined as follows.

If α∈N and (f,y)∈X, then α(f,y):=(αy(f),y),
If β∈H and (f,y)∈X, then β(f,y):=(f,β(y)).

Let G be the subgroup of Sym⁡(X) generated by the actions of N and of H on X; we view N and H as subsets of G. If α∈N and β∈H, then

β-1⁢α⁢β⁢(f,y)=β-1⁢α⁢(f,β⁢(y))=β-1⁢(αβ⁢(y)⁢(f),β⁢(y))=(αβ⁢(y)⁢(f),y),

so β-1⁢α⁢β∈N and N◁G. Then G=N⁢H and N∩H={idX}. Hence, the group G can be written as the semidirect product ∏y∈YA⋊H. The group G is called the wreath product of the groups A and H (with its canonical imprimitive action on X) and will be denoted by A≀H.

2.4 Interdefinability, bi-definability, bi-interpretability

We write A, B, C for the domains of the structures 𝔄, 𝔅, ℭ, respectively. If G is a set of permutations on a set A, then Inv⁡(G) denotes the relational structure 𝔄 with domain A which carries all relations that are preserved by all permutations of G. The operations Aut and Inv form a Galois connection between the set of all relational structures 𝔄 with domain A and the set of sets of permutations G on A (see, e.g., [9]). The permutation group Aut⁡(Inv⁡(G)) is the smallest permutation group containing G that is closed in Sym⁡(A) equipped with the topology of pointwise convergence. This topology is the restriction of the product topology on AA, where A is taken to be discrete. A permutation group G on A is closed in Sym⁡(A) if and only if G is the automorphism group of a relational structure. If 𝔄 is ω-categorical, then the structure Inv⁡(Aut⁡(𝔄)) is the expansion of 𝔄 by all relations that can be defined by a first-order formula in 𝔄 (this is a consequence of the proof of the theorem of Ryll-Nardzewski; see [45]).

It follows that Aut⁡(𝔄)⊆Aut⁡(𝔄′) if and only if all relations of 𝔄′ are first-order definable (without parameters) over 𝔄; in this case, we say that 𝔄′ is a first-order reduct of 𝔄. Two structures on the same domain are called interdefinable if they are reducts of one another. By the above, if 𝔄 or 𝔄′ is ω-categorical, then 𝔄 and 𝔄′ are interdefinable if and only if Aut⁡(𝔄)=Aut⁡(𝔄′).

Two structures 𝔄 and 𝔅, not necessarily with the same domain, are called bi-definable if there exists a bijection f:A→B between the domains of 𝔄 and 𝔅 such that 𝔄 and 𝔅 are interdefinable after identifying A and B along f. It follows that two ω-categorical structures 𝔄 and 𝔅 are bi-definable if and only if Aut⁡(𝔄) and Aut⁡(𝔄′) are isomorphic as permutation groups. For example, the structures (ℤ;{0}) and (ℤ;{1}) are bi-definable, but not interdefinable.

A (d-dimensional) interpretation of 𝔅 in 𝔄 is a partial surjective map I from Ad to B such that the pre-image of B, of the equality relation on B, and of each relation of 𝔅 under I is first-order definable in 𝔄. If 𝔄 has a d-dimensional first-order interpretation I in 𝔅 and 𝔅 has an e-dimensional first-order interpretation J in 𝔄 such that the relation

{(x,y1,1,…,yd,e)∣x=J⁢(I⁢(y1,1,…,yd,1),…,I⁢(y1,e,…,yd,e))}

is first-order definable in 𝔅 and

{(x,y1,1,…,yd,e)∣x=I⁢(J⁢(y1,1,…,y1,e),…,J⁢(yd,1,…,yd,e))}

is first-order definable in 𝔄, then 𝔄 and 𝔅 are called bi-interpretable. By a result of Coquand, Ahlbrandt, and Ziegler [3], two ω-categorical structures 𝔄 and 𝔅 are bi-interpretable if and only if Aut⁡(𝔄) and Aut⁡(𝔅) are topologically isomorphic, i.e., isomorphic via a mapping which is a homeomorphism with respect to the pointwise convergence topology.

2.5 Orbit growth and some classes of structures

Let X be a countably infinite set. There are three natural counting sequences attached to a permutation group on X, introduced and discussed in general in [31, 30].

Definition 2.2.

Let G⊆Sym⁡(X) be a permutation group, and let n∈ℕ. Then

on⁢(G) denotes the number of n-orbits of G, i.e., the number of orbits of the natural action G↷Xn;
oni⁢(G) denotes the number of injective n-orbits of G, i.e., the number of orbits of the natural action G↷X(n) where
X(n):={(x1,…,xn)∈Xn∣xi≠xjfor all distincti,j∈{1,…,n}};
ons⁢(G) denotes the number of orbits of n-subsets of G, i.e., the number of orbits of the natural action G↷(Xn) (={Y⊂Xn∣|Y|=n}).

If 𝔄 is a structure, then let

on(𝔄):=on(Aut(𝔄)),oni(𝔄):=oni(Aut(𝔄)),ons(𝔄):=ons(Aut(𝔄)).

In the notation above, we omit the reference to the group G or the structure 𝔄 if it is clear from the context.

A permutation group is called transitive if oi⁢(G)=1 and highly transitive if oi⁢(G)=1 for all i∈ℕ.

Definition 2.3.

A permutation group G⊆Sym⁡(X) is called oligomorphic if on⁢(G) is finite for all n.

Clearly, in Definition 2.3, we could have equivalently required that oni or ons are finite for all n. By the theorem of Engeler, Ryll-Nardzewski, and Svenonius, a countably infinite relational structure 𝔄 is ω-categorical if and only if Aut⁡(𝔄) is oligomorphic (see for instance [45]). In this paper, we are particularly interested in the following classes of structures and permutation groups.

Definition 2.4.

We introduce the following classes of permutation groups and structures.

Let 𝒢exp denote the class of those permutation groups G acting on a countable set X for which there is a constant c such that oni⁢(G)≤cn.
Let 𝒦exp denote the class of all countable structures 𝔄 with an automorphism group in 𝒢exp.
Let 𝒢exp+ denote the class of those permutation groups G acting on a countable set X for which there are constants c and d<1 such that oni⁢(G)≤c⁢nd⁢n.
Let 𝒦exp+ denote the class of all countable structures 𝔄 with an automorphism group in 𝒢exp+.

Remark 2.5.

Note that the conditions G∈𝒢exp and G∈𝒢exp+ imply that G is oligomorphic, and therefore 𝔄∈𝒦exp and 𝔄∈𝒦exp+ imply that 𝔄 is ω-categorical.

We write ℕ not only for the set of natural numbers, but also for the structure with the empty signature whose domain is ℕ.

Definition 2.6.

We write

𝒮 for the class of all at most countable structures that are first-order interdefinable with a structure having the empty signature,
𝒰 for the class of at most countable structures that are first-order interdefinable with a structure having a finite signature of unary relation symbols,
𝒰* for the class of the structures 𝔄∈𝒰 such that every orbit of Aut⁡(𝔄) is either a singleton or infinite.

When 𝒞 is a class of structures, we write 𝒞nf for the class consisting of all the structures in 𝒞 that have no finite orbits. Note that ℕ∈𝒮⊂𝒰nf⊂𝒰*⊂𝒰 and that (𝒰*)nf=𝒰nf.

2.6 Congruences of oligomorphic groups

We need the following easy observation about oligomorphic groups.

Proposition 2.7.

Every oligomorphic permutation group has finitely many congruences.

Proof.

Every congruence of a permutation group is a union of its 2-orbits. Then the claim follows directly from oligomorphicity. ∎

Lemma 2.8.

Let G be an oligomorphic permutation group, and let ∼ be a congruence of G which has finite equivalence classes. Then a∼b implies b∈aclG⁡(a).

Proof.

Suppose that a∼b, but b∉acl⁡(a). Then the orbit of b in Ga is infinite. Let b′ be any element in this orbit. Then, by definition, a∼b′. Hence, the equivalence class of a is infinite, a contradiction. ∎

If ∼1 and ∼2 are congruences, then the inclusion-wise smallest congruence relation that contains both ∼1 and ∼2 is called the equivalence relation generated by ∼1 and ∼2.

Lemma 2.9.

Let G be an oligomorphic permutation group, and let ∼1 and ∼2 be congruences of G with finite classes. Then the congruence generated by ∼1 and ∼2 also has finite classes.

Proof.

Let ∼ be the congruence generated by ∼1 and ∼2, and suppose that a∼b. Then there exists a sequence a0,b0,a1,b1,…,ak,bk with a0=a and bk=b such that ai∼1bi for all i≤k and bi∼2ai+1 for all i<k. By Lemma 2.8, this implies that bi∈acl⁡(ai) and ai+1∈acl⁡(bi) for all i. Since acl is a closure operator, it follows that b∈acl⁡(a). Since G is oligomorphic, it follows that acl⁡(a) is finite. Therefore, the equivalence class of a is also finite. ∎

Definition 2.10.

Let G be an oligomorphic permutation group. Then

∇⁡(G) denotes the intersection of all congruences of G with finitely many classes,
Δ⁢(G) denotes the smallest congruence that contains all congruences of G with finite classes.

If 𝔄 is an ω-categorical structure, then we use the notation ∇(𝔄):=∇(Aut(𝔄)), and Δ(𝔄):=Δ(Aut(𝔄)).

Remark 2.11.

Since G has finitely many congruences, it follows that ∇⁡(G) also has finitely many classes, i.e., it is the finest congruence of G with finitely many classes. By Lemma 2.9, it follows that every class of Δ⁢(G) is finite, i.e., Δ⁢(G) is the coarsest congruence of G with finite classes.

Remark 2.12.

If x and y are in the same orbit, then their Δ-classes have the same size. If G has finitely many orbits, it follows that there exists some n∈ℕ such that all elements lie in a Δ-class of size at most n.

The congruence Δ has the following equivalent description.

Lemma 2.13.

Let G be an oligomorphic permutation group on a countably infinite set X. Then (x,y)∈Δ⁢(G) if and only if y∈aclG⁡(x) and x∈aclG⁡(y).

Proof.

Let Δ′⁢(G)={(x,y)∣y∈acl⁡(x)∧x∈acl⁡(y)}. We claim that Δ′⁢(G) is an equivalence relation. It is clear that Δ′⁢(G) is reflexive and symmetric. The transitivity follows from the fact that acl is a closure operator. It is also clear from the definition that Δ′⁢(G) is preserved by G. Hence, Δ′⁢(G) is a congruence. For any x∈X, we have [x]Δ′⁢(G)⊆aclG⁡(x), so every class of Δ′⁢(G) is finite. Therefore, Δ′⁢(G) is finer than Δ⁢(G). On the other hand, if (x,y)∈Δ⁢(G), then y∈acl⁡(x) and x∈acl⁡(y), and thus (x,y)∈Δ⁢(G). ∎

We often use the following observation throughout this text.

Lemma 2.14.

Let G be an oligomorphic permutation group. Then every class of ∇⁡(G) is either infinite or a singleton.

Proof.

If the class of x∈X is finite, then its orbit is also finite. Indeed, let O be the orbit of x. Then every class of ∇⁡(G) in O is of the same size. So if this size is finite, then O is also finite since ∇ has finitely many classes.

Let Xfin be the union of the finite orbits of G. By oligomorphicity, it follows that Xfin is finite. Then ∇′:=∇(G)∩{(x,x)∣x∈Xfin} is also a congruence of ∇⁡(G). Since Xfin is finite, the congruence ∇′ has finitely many classes. This implies that ∇′=∇⁡(G), and thus every class of ∇⁡(G) within Xfin is a singleton. ∎

Lemma 2.15.

Let G be an oligomorphic permutation group on X, and let ∼ be a congruence of G with finite classes. Then the congruence generated by ∼ and ∇⁡(G) equals {(x,y)∈X2∣([x]∼,[y]∼)∈∇(G/∼)}.

Proof.

If π:X→X/∼ is the factor map x↦[x]∼, and ≈ is a congruence of G/∼, then

π-1(≈):={(x,y)∈X2∣(π(x),π(y))∈≈}

is a congruence of G which is coarser than ∼. In fact, π-1 defines a bijection between the congruences of G/∼ and those congruences of G which are coarser than ∼. The congruence π-1(∇(G/∼)) has finitely many classes since ∇(G/∼) has finitely many classes. Hence, π-1(∇(G/∼) is the finest congruence of G that is coarser than ∼ and has finitely many classes. So, by definition, it equals the congruence generated by ∼ and ∇⁡(G). ∎

2.7 Finite covers

We now introduce the concept of finite covers that plays a central role in this article. Forming finite covers may be viewed as a way to construct new ω-categorical structures from known ones; a more appropriate way is to view them as a way to decompose ω-categorical structures into (hopefully) simpler parts.

Definition 2.16.

Let 𝔄 and 𝔅 be structures. A mapping π:𝔄→𝔅 is called a finite covering map (or finite cover) if

π is surjective,
for each w∈B, the set π-1⁢(w) is finite,
∼π is preserved by Aut⁡(𝔄),
the image of Aut⁡(𝔄) under μπ equals Aut⁡(𝔅).

(See Definition 2.1 for the definition of ∼π and μπ.) The sets π-1⁢(w), for w∈B, are called the fibers of the finite covering map π. A structure 𝔄 is called a finite covering structure of 𝔅 if there is a finite covering map π:𝔄→𝔅.

Remark 2.17.

A finite covering structure of an ω-categorical structure has an oligomorphic automorphism group, and hence is ω-categorical.

Remark 2.18.

Let 𝔄 be an arbitrary structure and ∼ a congruence of Aut⁡(𝔄). If all ∼-classes are finite, then 𝔄 is a finite covering structure of the quotient structure 𝔄/∼, where 𝔄/∼ can be any structure such that Aut(𝔄/∼)=Aut(𝔄)/∼. In fact, every finite covering structure is of this form. Indeed, let 𝔄 be a structure. If π:𝔄→𝔅 is a finite covering map, then ∼π is a congruence of Aut⁡(𝔄) and there is a natural bijection between B and A/∼π defined by w↦π-1⁢(w). Let us identify B and of A/∼π along this bijection, and let 𝔄/∼π be any structure such that Aut(𝔄/∼π)=Aut(𝔅). The image of Aut⁡(𝔄) under the homomorphism μπ equals Aut⁡(𝔅); hence Aut(𝔄)/∼π=Aut(𝔅)=Aut(𝔄/∼π).

We present a series of simple examples of finite covers; they illustrate different phenomena of finite covers on which we will comment later, referring back to these examples.

Example 2.19.

Let P→1⋅ω be the directed graph which is an infinite union of directed edges. Then P→1⋅ω is a finite covering structure of ℕ, with π being the projection to the second argument. Also note that Aut⁡(P→1⋅ω) is topologically isomorphic to Sym⁡(ℕ), and that P→1⋅ω and (ℕ;≠) are bi-interpretable but not bi-definable.

Example 2.20.

Let K2⋅ω be the graph which is an infinite union of undirected edges. Then K2⋅ω is a finite covering structure of ℕ, with π being the projection to the second argument. Identifying the domain of K2⋅ω with {0,1}×ℕ so that (u,n) is adjacent to (v,m) if and only if n=m and u≠v, the automorphism group of K2⋅ω is the wreath product ℤ2≀Sym⁡(ω) (see Section 2.3).

Example 2.21.

Let K2⋅ω be the structure with domain {0,1}×ℕ from Example 2.20, and let 𝔄 be the expansion of K2⋅ω by the equivalence relation Eq defined by Eq⁡((u,n),(v,m)) if and only if u=v. Then 𝔄 is a finite covering structure of ℕ with respect to the covering map π that maps (u,n) to n. Note that Aut⁡(𝔄) is isomorphic (as an abstract group) to the direct product ℤ2×Sym⁡(ℕ) (see Section 2.2 for direct products and other actions of direct products).

Example 2.22.

Let A:={0,1,2,3}×ℕ, and let π:A→ℕ be the projection to the second argument. Let 𝔄 be the graph with vertex set A such that (u,b) is adjacent to (v,c) if and only if

b=c and u=v+1mod4, or
b≠c and u=vmod2.

See Figure 1 for an illustration. Note that 𝔄 is a finite covering structure of ℕ with respect to π. The automorphism group of 𝔄 equals KH, where

H={α∈Sym⁡(A)∣if⁢α⁢(u,v)=(u′,v′),then⁢u=u′} (i.e., H is topologically isomorphic to Sym⁡(ℕ)), and
K={α∈∏i∈ℕℤ4∣for all⁢k,l∈ℕ,αk⁢ℤ2=αl⁢ℤ2}, where ℤk is the cyclic group acting on {0,1,…,k-1} and ∏i∈ℕℤ4 is the direct product in its intransitive action on A (see Section 2.2).

$Figure 1 An illustration of the subgraph of the structure 𝔄{\mathfrak{A}} from Example 2.22 that is induced by 2 fibers.$

Figure 1

An illustration of the subgraph of the structure 𝔄 from Example 2.22 that is induced by 2 fibers.

Example 2.23.

Let 𝔅 be the countable structure which carries an equivalence relation Eq with three classes R,S,T such that |S|=|T|, and a unary relation symbol denoting the class R. Let A:=({0,1}×R)∪({0}×(S∪T)). We define the structure 𝔄 with domain A and the signature {E,F}, where E and F have arity two, and

E⁢((u1,b1),(u2,b2)) holds if and only if (u1=0, b1∈R, b2∈S) or (u1=1, b1∈R, b2∈T),
F⁢((u1,b1),(u2,b2)) holds if and only if b1=b2.

See Figure 2. Let π:A→B be the projection to the second argument. Then we have ∼π=F, and π is a finite covering. If R,S,T are countably infinite, then ∼π=F=Δ(𝔄). The automorphism group of 𝔄 is isomorphic to a semidirect product (Sym(R)×Sym(S)2)⋊ℤ2.

$Figure 2 An illustration of a subgraph of the structure 𝔄{\mathfrak{A}} from Example 2.23 for the special case |S|=|T|=1{\lvert S\rvert=\lvert T\rvert=1}.$

Figure 2

An illustration of a subgraph of the structure 𝔄 from Example 2.23 for the special case |S|=|T|=1.

Definition 2.24.

Let π:𝔄→𝔅 be a finite covering map, b∈B, and S:=π-1(b).

The fiber group of π at b is the group Aut(𝔄){S}|S.
The binding group of π at b is the group K|S, where K is the kernel of μπ.

So the binding group at b is a normal subgroup of the fiber group at b. If, for some b∈B, the fiber group and the binding group at b are unequal, then π is called twisted. Example 2.23 gives an example of a twisted finite cover; Examples 2.19, 2.20, 2.21, and 2.22 are not twisted.

Remark 2.25.

The following terminology is not needed for stating or proving our results, but we mention it for a better understanding of the examples of finite covers that we have already presented. Let π:𝔄→𝔅 be a finite covering map, and let Bb be the binding group at b∈B. Then π is called free if the kernel of μπ:Aut⁡(𝔄)→Aut⁡(𝔅) equals ∏b∈BBb. Example 2.20, Example 2.19, Example 2.22, and Example 2.23 are free. Example 2.21 is an example of a finite cover which is not free: the binding group at each point is ℤ2 and equals the kernel of μπ:Aut⁡(𝔄)→Aut⁡(𝔅), which is therefore not equal to ∏b∈BBb=ℤ2ω.

2.8 Trivial finite covers

There are two important notions of triviality for finite covers, intended to describe those finite covers that have an automorphism group which is smallest possible. This is important for our purposes since we will describe general finite covering structures in our class by describing them as certain first-order reducts of trivial finite covers, and as we will see, trivial covers are much easier to describe.

Definition 2.26.

Let π:𝔄→𝔅 be a finite covering map. We say that π is

a trivial cover if the kernel of μπ:Aut⁡(𝔄)→Aut⁡(𝔅) is trivial (only contains the identity permutation idA),
a strongly trivial cover if all of its fiber groups are trivial.

A structure 𝔄 is called a (strongly) trivial covering structure of 𝔅 if there is a finite covering map π:𝔄→𝔅 which is (strongly) trivial.

It is clear from the definition that π is a trivial cover if and only if all of its binding groups are trivial. Hence, if π is strongly trivial, then it is also trivial. Example 2.19 is an example of a strongly trivial finite covering. Example 2.23 is an example of a trivial finite covering which is not a strongly trivial finite covering. Examples 2.20, 2.21, and 2.22 are examples of nontrivial finite coverings.

Next we give a sufficient condition for a structure 𝔅 under which every trivial cover of 𝔅 is strongly trivial.

Lemma 2.27.

Let B be a structure such that the stabiliser Aut(B)b for every b∈B has no nontrivial finite-index subgroups. Then every trivial cover of B is strongly trivial.

Proof.

Let π:𝔄→𝔅 be a trivial finite cover. Then μπ is an isomorphism between Aut⁡(𝔄) and Aut⁡(𝔅). Let b∈B. We need to show that the fiber group of π at b is trivial. Put S:=π-1(b). Let us consider the mapping φ:Aut(𝔅)b→Sym(S) given by h↦μπ-1⁢(h)|S. Then φ is clearly a group homomorphism. Let K be the kernel of this homomorphism. Then K is a finite index subgroup of Aut(𝔅)b, and thus, by our assumption, K=Aut(𝔅)b. That is, φ is the trivial homomorphism, which means that the fiber group of π at b is trivial. ∎

We now give an explicit description of strongly trivial covers.

Lemma 2.28.

Let π:A→B be a strongly trivial covering map. Then, for each orbit O of B, there exists a finite set FO and a mapping ψO:π-1⁢(O)→FO such that,

for every w∈O, the restriction of ψO to π-1⁢(w) is a bijection,
ψO⁢(x)=ψO⁢(μπ-1⁢(β)⁢(x)) for all x∈π-1⁢(O) and β∈Aut⁡(𝔅).

Proof.

Let us fix an element b∈O, and let FO:=π-1(b). If x∈π-1⁢(O), then there exists an automorphism g of 𝔅 such that g⁢(π⁢(x))=b. Let us define ψO⁢(x) to be μπ-1⁢(g)⁢(x). We claim that ψO⁢(x)∈FO and that its value is well-defined (i.e., it does not depend on our particular choice of g). The first claim is clear since, by definition,

π⁢(μπ-1⁢(g)⁢(x))=μπ⁢(μπ-1⁢(g))⁢(π⁢(x))=g⁢(π⁢(x))=b;

thus μπ-1⁢(g)⁢(x)∈π-1⁢(b)=FO.

In order to show the second claim, we need to show that if h∈Aut⁡(𝔅) is such that h⁢(π⁢(x))=g⁢(π⁢(x))=b, then

μπ-1⁢(g)⁢(x)=μπ-1⁢(h)⁢(x).

Since (h-1⁢g)⁢(π⁢(x))=π⁢(x), it follows that (μπ-1⁢(h-1⁢g))|π⁢(x) is in the fiber group at π⁢(x). Since π is strongly trivial, this group is trivial, and hence

μπ-1⁢(h-1⁢g)⁢(x)=x.

This implies that

μπ-1⁢(h)⁢(x)=(μπ-1⁢(h)⁢μπ-1⁢(h-1⁢g))⁢(x)=μπ-1⁢(g)⁢(x).

Now the first item follows from the fact that if w∈O is such that g⁢(w)=b, then g defines a bijection between π-1⁢(w) and π-1⁢(b). As for the second item, let x∈π-1⁢(O), and let g∈Aut⁡(𝔅) be such that g⁢(π⁢(x))=b. If β∈Aut⁡(𝔅), then

(g⁢β-1)⁢(π⁢(μπ-1⁢(β)⁢(x)))=(g⁢β-1)⁢(β⁢(π⁢(x)))=g⁢(π⁢(x))=b,

and thus

ψO⁢(μπ-1⁢(β)⁢(x))=(μπ-1⁢(g⁢β-1))⁢(μπ-1⁢(β)⁢(x))=μπ-1⁢(g)⁢(x)=ψO⁢(x).∎

Remark 2.29.

Let the sets FO and the maps ψO be defined as in Lemma 2.28 for each orbit O of Aut⁡(𝔅). Then there is a natural bijection between A and ⋃O(FO×O) defined as x↦(ψO⁢(x),π⁢(x)), where O is the orbit of Aut⁡(𝔅) containing π⁢(x). If we identify each element of A with its image under this bijection, then Aut⁡(𝔄) consists of those permutations that fix the first coordinate of each element and that act as an automorphism of 𝔅 on the second coordinate.

2.9 Covering reducts

Let 𝔄 and 𝔅 be structures, and let π:𝔄→𝔅 be a finite covering map. A first-order reduct ℭ of 𝔄 is a covering reduct of 𝔄 with respect to π (and 𝔄 is called a covering expansion of ℭ with respect to π; see [39]) if every α∈Aut⁡(ℭ) preserves ∼π and μπ⁢(α)∈Aut⁡(𝔅).

Remark 2.30.

We do not need but mention that every finite cover π:𝔄→𝔅 is a covering expansion of a free finite covering structure of 𝔅 with respect to π ([39, Lemma 2.1.3]).

Definition 2.31.

Let π:𝔄→𝔅 be a finite covering map.

If 𝔄 is a covering reduct of a trivial covering structure of 𝔅 with respect to π, then π is called a split cover of 𝔅 [39] (in this case, we also say that π is split).
If 𝔄 is a covering reduct of a strongly trivial covering of 𝔅 with respect to π, then π is called a strongly split cover of 𝔅 [39].

Equivalently (this motivates the terminology; see [39]), a finite cover π:𝔄→𝔅 is split if the kernel K of μπ:Aut⁡(𝔄)→Aut⁡(𝔅) has a closed complement in Aut⁡(𝔄), i.e., there is a closed subgroup H of Aut⁡(𝔄) such that K⁢H=Aut⁡(𝔄) and K∩H={1} (so that Aut⁡(𝔄) is isomorphic to the semidirect product K⋊H). Examples 2.19, 2.20 2.21, and 2.22 are examples of split covers of ℕ. For a non-example, see, e.g., [40]. Example 2.23, in the case that |S|=|T|=1, is an example of a finite split cover of a structure in 𝒰 which is not strongly split.

2.10 Operations on classes of structures

Let 𝔄 be a structure, and let 𝔅 be a first-order reduct of 𝔄. Then we say that 𝔅 is a finite index (first-order) reduct of 𝔄 if and only if the index [Aut⁡(𝔅):Aut⁡(𝔄)] is finite. We define the following operations on classes of structures.

Definition 2.32.

Let 𝔄 be a countable ω-categorical structure. Then

C⁢(𝔄) is the class of structures which are interdefinable with an expansion of 𝔄 with finitely many constants,
M⁢(𝔄) is the class of structures that are interdefinable with the (up to isomorphism unique [8, 12]) model-complete core of 𝔄,
R⁢(𝔄) is the class of first-order reducts of 𝔄,
R<∞⁢(𝔄) is the class of finite index first-order reducts of 𝔄,
F⁢(𝔄) is the class of finite covering structures of 𝔄.

If 𝒞 is a class of structures and Φ is one of the operators above, then we use the notation Φ⁢(𝒞) for the union of the classes Φ⁢(𝔄) such that 𝔄∈𝒞.

Proposition 2.33.

The following identities hold.

C∘C=C,
M∘M=M,
R∘R=R,
R<∞∘R<∞=R<∞,
F∘F=F,
C⁢(𝒰nf)=𝒰*,
R⁢(𝒰)=R⁢(𝒰*),
𝒦exp=R⁢(𝒦exp),
𝒦exp+=R⁢(𝒦exp+).

Proof.

Straightforward from the definitions. ∎

We will show that 𝒦exp=R⁢(𝒰) and 𝒦exp+=(F∘R)⁢(𝒰)=(R∘F)⁢(𝒰), and we will give several equivalent descriptions of these classes in Section 8. We also prove Thomas’ conjecture for each structure in 𝒦exp+ (Theorem 6.37).

3 Reducts of unary structures

In this section, we characterise first-order reducts of unary structures in terms of their automorphism groups, and in particular prove Thomas’ conjecture for the class R⁢(𝒰). We mention that the finite-domain constraint satisfaction tractability conjecture has been shown for all structures in R⁢(𝒰) (see [15]).

Lemma 3.1.

Let A be a structure. Then A∈U if and only if there are finitely many sets O1,…,Ok such that Aut⁡(A)=∏i=1kSym⁡(Oi).

Proof.

Suppose that 𝔄∈𝒰. Then 𝔄 is interdefinable with a unary structure 𝔄′; let O1,…,Ok be the minimal non-empty intersections of predicates from 𝔄′; clearly, these sets partition A. The containment Aut⁡(𝔄)⊆∏i=1kSym⁡(Oi) is clear since every automorphism of 𝔄 is an automorphism of 𝔄′ and hence preserves the sets O1,…,Ok. For the reverse containment, let α∈Sym⁡(A) be such that α⁢(Oi)=Oi for all i≤k. Since Aut⁡(𝔄)=Aut⁡(𝔄′), we need to show that α preserves all (unary) relations U of 𝔄′. Let x∈U, and let i≤k be such that x∈Oi. Since U must be a union of orbits of Aut⁡(𝔄′), we have that α⁢(x)∈Oi also lies in U.

Conversely, suppose that Aut⁡(𝔄)=∏i=1kSym⁡(Oi). Then 𝔄 is first-order interdefinable with the unary structure 𝔄=(X;O1,…,Ok). ∎

The following statement is an easy consequence of Lemmas 3.1 and 2.14.

Corollary 3.2.

Let A∈U. Then the ∇⁡(A)-classes are the infinite orbits of Aut⁡(A) and the singleton orbits.

Lemma 3.3.

Let A∈R⁢(U), and let C1,…,Ck be the ∇⁡(A)-classes. Then

∏i=1kSym⁡(Ci)⊆Aut⁡(𝔄).

Proof.

Let 𝔄 be a first-order reduct of a structure 𝔅∈𝒰. Let O1,…,Ol be the orbits of 𝔅. Then Aut⁡(𝔅)=∏i=1lSym⁡(Oi) by Lemma 3.1. Let us define the binary relation R on A so that xRy if and only if x=y or the transposition (x⁢y) is contained in Aut⁡(𝔄). Then it is easy to see that R is a congruence of Aut⁡(𝔄). On the other hand, Aut⁡(𝔄)⊇Aut⁡(𝔅)=∏i=1lSym⁡(Oi) implies that each class of R is the union of some of the orbits of Aut⁡(𝔅). In particular, R has finitely many classes, and so, by definition, the congruence ∇⁡(𝔄) is finer that R. This means that, for all x,y∈Ci, the transposition (x⁢y) is contained in Aut⁡(𝔄). Therefore, ∏i=1kSym⁡(Ci)⊆Aut⁡(𝔄) since Aut⁡(𝔄) is closed. ∎

Corollary 3.4.

R⁢(𝒰)=R<∞⁢(𝒰)=R<∞⁢(𝒰*).

Proof.

The containments “⊇” are obvious. Let 𝔄∈R⁢(𝒰). Let C1,…,Ck be the classes of ∇⁡(𝔄). Then the group Aut⁡(𝔄) acts on the set {C1,…,Ck}. By Lemma 3.3, the kernel of this action is ∏i=1kSym⁡(Ci). In particular, the index of ∏i=1kSym⁡(Ci) in Aut⁡(𝔄) is finite. On the other hand, ∏i=1kSym⁡(Ci) is the automorphism group of a unary ω-categorical structure with orbits C1,…,Ck. We also know from Lemma 2.14 that each class Ci is either a singleton or infinite. Therefore, 𝔄∈R<∞⁢(𝒰*). ∎

The following has been shown in [15, Proposition 6.8]; the proof we present here is simpler.

Corollary 3.5.

Let A∈R⁢(U). Then there exists an expansion of A with finitely many constants which is in U*.

Proof.

Let C1,…,Ck be the classes of ∇⁡(𝔄), and let us choose elements ci∈Ci. We claim that the structure (𝔄,c1,…,ck) is in 𝒰*. By Lemma 2.14, we know that each class Ci is either a singleton or infinite. Without loss of generality, we can assume that Ci={ci} for i=1,…,l and Cj is infinite for j>l. We claim that

Aut⁡(𝔄;c1,…,ck)=∏i=1kid⁡({ci})×∏i=l+1kSym⁡(Ci∖{ci}).

Then Lemma 3.1 implies that (𝔄,c1,…,ck)∈𝒰*. To prove the claim, first recall from Lemma 3.3 that ∏i=1kSym⁡(Ci)⊆Aut⁡(𝔄), and hence

∏i=1kSym(Ci){c1,…,ck}⊆Aut(𝔄){c1,…,ck}.

Since every automorphism of 𝔄 that fixes c1,…,ck must also preserve the sets C1,…,Ck, we in fact have equality ∏i=1kSym(Ci){c1,…,ck}=Aut(𝔄){c1,…,ck}. Thus,

Aut⁡(𝔄;c1,…,ck)=Aut(𝔄){c1,…,ck}=∏i=1kSym(Ci){c1,…,ck}=∏i=1kid⁡({ci})×∏i=l+1kSym⁡(Ci∖{ci}).∎

Lemma 3.6.

Let A∈R⁢(U), and let C1,…,Ck be the ∇⁡(A)-classes. Then we have Aut⁡(A)=∏i=1kSym⁡(Ci)⋊A, where A is a subgroup of Aut⁡(A) acting faithfully on {C1,…,Ck}.

Proof.

Recall from Lemma 2.14 that every class of ∇⁡(𝔄) is either infinite or a singleton. We can assume that C1,…,Cl are infinite and, for every j∈{l+1,…,k}, there exists cj∈A such that Cj={cj}. Let ei:Ci→ℕ, for 1≤j≤l, be bijections, and let e=⋃i∈{1,…,l}ei. Then it is easy to see that, for every α∈Aut⁡(𝔄), there exists α~∈Sym⁡(A) such that

α and α~ have the same action on the set {C1,…,Ck},
e⁢(α~⁢(x))=e⁢(x) for every x∈⋃i∈{1,…,l}Ci.

The permutation α-1⁢α~ fixes every class of ∇⁡(𝔄). Since

N:=∏i=1kSym(Ci)⊆Aut(𝔄)

by Lemma 3.3, it follows that α-1⁢α~∈Aut⁡(𝔄) and thus α~∈Aut⁡(𝔄). Let

A:={α~∣α∈Aut(𝔄)}.

Then A⊆Aut⁡(𝔄) is a subgroup of G which acts faithfully on {C1,…,Ck}. Then it is also clear that N is a normal subgroup of Aut⁡(𝔄) since it is the kernel of the action of Aut⁡(𝔄) on {C1,…,Ck}. Therefore, Aut⁡(𝔄) can be written as a semidirect product, Aut⁡(𝔄)=∏i=1kSym⁡(Ci)⋊A. ∎

Corollary 3.7.

Let A∈R⁢(U) be with no finite orbits. Then Aut⁡(A) is isomorphic to the wreath product Sym⁡(N)≀A for some permutation group A on a finite set.

Proof.

Let C1,…,Ck be the classes of ∇⁡(𝔄). Without loss of generality, we can assume that Ci={(i,n)∣n∈ℕ}. Let A be the image of the action of Aut⁡(𝔄) on the set {C1,…,Ck}. Then if we use the bijections ei:(i,n)↦n in the proof of Lemma 3.6, the statement of the corollary follows. ∎

Lemma 3.8.

Suppose that A,B∈R⁢(U) have the same domain. If ∇⁡(A)=∇⁡(B) and the actions of the groups Aut⁡(A) and Aut⁡(B) on the ∇⁡(A)-classes are the same, then A and B are interdefinable.

Proof.

By the ω-categoricity of 𝔄 and 𝔅, it is enough to show that

Aut⁡(𝔄)=Aut⁡(𝔅).

Let C1,…,Ck be the classes of ∇⁡(𝔄)=∇⁡(𝔅). Lemma 3.3 shows that

∏i=1kSym⁡(Ci)⊆Aut⁡(𝔄).

Now let β∈Aut⁡(𝔅). By our assumption about the action of Aut⁡(𝔄) and Aut⁡(𝔅) on the ∇⁡(𝔄)-classes, there is a permutation α∈Aut⁡(𝔄) such that β⁢α-1 fixes each class Ci. Then β⁢α-1∈∏i=1kSym⁡(Ci)⊆Aut⁡(𝔄), and so β∈Aut⁡(𝔄). Therefore, Aut⁡(𝔅)⊆Aut⁡(𝔄). Analogously, Aut⁡(𝔄)⊆Aut⁡(𝔅). ∎

Corollary 3.9.

Every structure in U has finitely many first-order reducts.

Proof.

Let 𝔄∈𝒰. Then, by Lemma 3.1, Aut⁡(𝔄)=∏i=1kSym⁡(Oi). If 𝔅 is a first-order reduct of 𝔄, then ∇⁡(𝔅) is a union of orbits of 𝔄. This means that there are finitely many choices for the relation ∇⁡(𝔅). If the relation ∇⁡(𝔅) is fixed, then there are finitely many possible actions of Aut⁡(𝔅) on the classes of ∇⁡(𝔅). By Lemma 3.8, it follows that ∇⁡(𝔅) and the action of Aut⁡(𝔅) on the classes of ∇⁡(𝔅) already determine the structure 𝔅 up to interdefinability. Therefore, 𝔄 has finitely many first-order reducts. ∎

We also obtain an equivalent description of first-order reducts of unary structures in terms of their automorphism groups.

Corollary 3.10.

A structure A is in R⁢(U) if and only if ∏i=1kSym⁡(Ci)⊆Aut⁡(A) for some partition of A into classes C1,…,Ck.

Proof.

One direction has been shown in Lemma 3.3. For the converse implication, suppose that Aut⁡(𝔄) contains ∏i=1kSym⁡(Ci) for some partition of A into classes C1,…,Ck. Note that Aut⁡(A;C1,…,Ck)=∏i=1kSym⁡(Ci) (see Lemma 3.1) and that 𝔄 is a first-order reduct of (A;C1,…,Ck). Hence, 𝔄∈R⁢(𝒰). ∎

4 Finite coverings of unary structures

In this section, we classify the finite coverings of unary structures. First we make the following observation.

Lemma 4.1.

F⁢(𝒰)=F⁢(𝒰*).

Proof.

The containment “⊇” is trivial. In order to show the other direction, it is enough to show that 𝒰⊆F⁢(𝒰*) since F∘F=F. So let 𝔄∈𝒰, and let F be the union of the finite orbits of 𝔄. Then F is finite. Let us consider the unary structure 𝔅 whose domain is B:=A∖F∪{x} for any x∉A, and whose relations are the infinite orbits of 𝔄 and {x}. Then 𝔅∈𝒰*. Let π:𝔄→𝔅 be defined as π⁢(y)=x if x∈F, and π⁢(y)=y otherwise. Then it is easy to see that π is a finite covering map, and hence 𝔄∈F⁢(𝒰*). ∎

The following theorem summarises the results from Section 4.1 and Section 4.2.

Theorem 4.2.

Let B∈U*, and let π:A→B be a finite covering map. Then A has finitely many covering reducts with respect to π.

Proof.

Proposition 4.9 shows that π is strongly split. The statement then follows from Corollary 4.28. ∎

4.1 Finite covers of unary structures split

The following series of lemmas is needed to show that every finite covering map of a structure 𝔅∈𝒰* is strongly split (Proposition 4.9). Throughout this subsection, let 𝔅∈𝒰*, and let π:𝔄→𝔅 be a finite covering map.

Remark 4.3.

Observe that 𝔅 satisfies the condition of Lemma 2.27, i.e., Aut(𝔅)x has no finite index subgroup for any x∈B. By Lemma 2.27, this implies that every trivial finite cover of 𝔅 is strongly trivial, and hence every split cover of 𝔅 is strongly split.

Remark 4.4.

When 𝔅 is taken from 𝒰 instead of 𝒰*, then there are split covers of 𝔅 that are not strongly split, as illustrated by Example 2.23 if |S|=|T|=1.

Lemma 4.5.

Let F be a finite subset of an infinite orbit O of B. If |F| is large enough, then there exists an automorphism α of A such that

α⁢(x)=x for all x∈E:=A∖π-1(F),
μπ⁢(α)|F is nontrivial.

Proof.

Let k be the maximum of the sizes of the fibers of π, and let p>k be a prime number. We claim that if |F|≥p, then there is an automorphism α of 𝔄 satisfying conditions (1) and (2).

Let u1,…,up∈F be distinct elements. Then the p-cycle (u1⁢u2⁢…⁢up) is contained in Aut⁡(𝔅) by Lemma 3.1. By the definition of finite covering maps, there exists β∈Aut⁡(𝔄) such that μπ⁢(β)=(u1⁢…⁢up). Now let α:=βk!∈Aut(𝔄). Then μπ⁢(α)|F is again a p-cycle and hence nontrivial. On the other hand, if u∈B∖F and U:=π-1(u), then β⁢(U)=U, and α|U=βk!|U=idU since |U|≤k. This means that α|E=idE. Therefore, α∈Aut⁡(𝔄) satisfies conditions (1) and (2), which proves the lemma. ∎

Recall that, for any finite set F of cardinality at least 5, the alternating group Alt⁡(F) is the only nontrivial proper normal subgroup of Sym⁡(F) (see, e.g., [36, Chapter 8.1]).

Lemma 4.6.

Let F be a finite subset of an infinite orbit O of B. If |F| is large enough, then for any pairwise distinct u1,u2,u3,u4∈F, there exists an automorphism α of A such that

α⁢(x)=x for all x∈E:=A∖π-1(F),
μπ⁢(α)|F=(u1⁢u2)⁢(u3⁢u4).

Proof.

Let K:={μπ(γ)|F∈Sym(F)∣γ∈Aut(𝔄)E}. We claim that K is a normal subgroup of Sym⁡(F). It is clear that K is a subgroup of Sym⁡(F). Let α∈K and β∈Sym⁡(F). We need to show that β⁢α⁢β-1∈K. By the definition of K, there exists γ∈Aut(𝔄)E so that μπ⁢(γ)|F=α. By Lemma 3.1, there exists δ∈Aut⁡(𝔅) so that δ|F=β. By the definition of finite covers, there exists η∈Aut⁡(𝔄) such that μπ⁢(η)=δ. Let γ′=η⁢γ⁢η-1∈Aut⁡(𝔄). Then on can check that γ′⁢(x)=x for all x∈E and

μπ⁢(γ′)|F=(μπ⁢(η)⁢μπ⁢(γ)⁢μπ⁢(η)-1)|F=δ|F⁢μπ⁢(γ)|F⁢δ-1|F=β⁢α⁢β-1.

We obtained that K◁Sym⁡(F). By Lemma 4.5, we know that if F is large enough, then K is nontrivial. Therefore, if F is large enough, then K≥Alt⁡(F), and then the statement of the lemma follows. ∎

Lemma 4.7.

Let O be an infinite orbit of B. Then, for all distinct v1,v2∈O, there exists an α∈Aut⁡(A) such that

α⁢(x)=x for all x∈A∖π-1⁢({v1,v2}),
μπ⁢(α)=(v1⁢v2).

Proof.

Let v1,v2∈O, and let us choose a finite subset F of O which contains the elements v1 and v2, which is large enough so that we can apply Lemma 4.6 for F. Choose u3,u4∈F such that u1:=v1,u2:=v2,u3,u4 are pairwise distinct. Let α∈Aut⁡(𝔄) be as in Lemma 4.6. For each i∈ℕ, choose γi∈Aut⁡(𝔄) so that

μπ⁢(γi)⁢(v1)=v2,

μπ⁢(γi)⁢(v2)=v1,

μπ⁢(γi)⁢(F)∩μπ⁢(γj)⁢(F)={v1,v2} for all⁢i≠j.

By Lemma 3.1, it follows that such γi’s exist. Let βi:=γiαγi-1∈Aut(𝔄). Then μπ⁢(βi)⁢(v1)=v2, μπ⁢(βi)⁢(v2)=v1, and for all x∈S:=π-1({v1,v2}), we have βi⁢(x)=x if i is large enough. Since there are finitely many possible actions of βi on the finite set S, there is a subsequence (βl⁢(i))i of (βi)i so that βl⁢(i)|S=βl⁢(j)|S for all i,j∈ℕ. Then the sequence βl⁢(i) converges to a permutation β for which β⁢(x)=x for all x∈A∖S and π⁢(β)=π⁢(βl⁢(1))=(v1⁢v2). Since Aut⁡(𝔄) is closed, it follows that β∈Aut⁡(𝔄), which finishes the proof of the lemma. ∎

Lemma 4.8.

Let O be the set of orbits of B. Then, for each O∈O, there exists a finite set FO and a mapping ψO:π-1⁢(O)→FO such that ψO|π-1⁢(y) is bijective for every y∈O, and Aut⁡(A) contains every α∈Sym⁡(A) such that

α preserves ∼π,
μπ⁢(α)∈Aut⁡(𝔅),
ψO⁢(x)=ψO⁢(α⁢(x)) for every O∈𝒪 and x∈π-1⁢(O).

Proof.

If O is finite, then O={u} for some u∈B since 𝔅∈𝒰*. In this case, let ψO=idπ-1⁢(u). If O is infinite, then we define ψO as follows. Let u∈O be arbitrary.

If x∈π-1⁢(u), then set ψO(x):=x.
If x∈π-1⁢(O)∖π-1⁢(u), then by Lemma 4.7, there exists a permutation
α∈Aut⁡(𝔄)|A∖π-1⁢({π⁢(x),u})
such that α⁢(π-1⁢(π⁢(x)))=π-1⁢(u). In particular, α defines a bijection between π-1(π(x))) and π-1⁢(u). Set ψO(x):=α(x).

We claim that these mappings satisfy the conditions of the lemma. Let G be the permutation group of those γ∈Sym⁡(B) for which there exists an automorphism α of 𝔄 with (μπ⁢(α))=γ and satisfying conditions (1)–(3) of the lemma. Then, since Aut⁡(𝔅)=∏O∈𝒪Sym⁡(O), it is enough to show that (GB∖O)|O=Sym⁡(O) for all O∈𝒪. If O is a singleton, then the claim is trivial, so we can assume that O is infinite. It is easy to see that G is closed. Thus, (GB∖O)|O is also closed. Hence, by Lemma 3.1, it is enough to show that G contains for all u1,u2∈O the transposition (u1⁢u2). For this, it is enough to show that (u⁢v)∈G for all v∈O∖{u}, where u is the element of O which is used in the definition of ψO. But this follows directly from the definition of the mapping ψO. ∎

Proposition 4.9.

Any finite covering map π:A→B for B∈U* is strongly split.

Proof.

Let FO and ψO be defined as in Lemma 4.8 for each orbit O of 𝔅. Let F:=⊔OFO and ψ:=⋃OψO. Let 𝔄′ be the expansion of 𝔄 obtained by adding to 𝔄 for each x∈F the unary relation ψ-1⁢(x). Then, by Lemma 4.8, it follows that μπ⁢(Aut⁡(𝔄′))=Aut⁡(𝔅)=μπ⁢(Aut⁡(𝔄)). Thus, 𝔄 is a covering reduct of 𝔄′. We claim that π:𝔄′→𝔅 is a strongly trivial cover. This implies the statement of the proposition. By Remark 4.3, it is enough to show that the finite cover π:𝔄′→𝔅 is trivial, i.e., that the kernel of the map μπ is trivial. Let α∈Aut⁡(𝔄′) be so that μπ⁢(α)=idB, and let x∈A. Then x∼πα⁢(x) and ψ⁢(x)=ψ⁢(α⁢(x)). It follows from the definition that ψ is injective on [x]π. This implies that x=α⁢(x) and hence that α=idA. Therefore, the kernel of μπ is trivial. ∎

Remark 4.10.

Proposition 4.9 generalises [67, Theorem 2.4], which states that every finite covering of ℕ strongly splits.

4.2 Covering reducts of trivial coverings

In this subsection, we describe the automorphism groups of covering reducts of a trivial finite covering of a structure in 𝒰*. In particular, we show that there are always finitely many of them.

Remark 4.11.

Let 𝔄 be a strongly trivial covering of a unary structure 𝔅 with orbits O1,…,Ok. Then, as in Remark 2.29, the elements of the structure 𝔄 can be identified with the elements of ⊔i=1k(Fi×Oi) for some finite sets Fi so that Aut⁡(𝔄) consists of exactly those permutations which preserve the first coordinate and stabilise the sets Oi in the second coordinate. In this case, Aut⁡(𝔄) can be written as ∏i∈{1,…,k}{idFi}≀Sym⁡(Oi). For convenience, we will always assume that the sets Fi are pairwise disjoint.

Remark 4.12.

It follows from the description of strongly trivial coverings of unary structures in Remark 4.11 that every (reduct of a) strongly trivial covering structure of a structure from 𝒰 has a first-order interpretation over (ℕ;=).

Throughout this subsection, let us fix a structure 𝔅∈𝒰* and a trivial finite covering map π:𝔄→𝔅. Let O1,…,Ok be the orbits of 𝔅. We identify the elements of the structure 𝔄 with the elements of ⊔i=1k(Fi×Oi) for some disjoint finite sets F1,…,Fk as explained in Remark 4.11, and define F:=⊔i=1kFi.

Definition 4.13.

We define two permutation groups.

Let S be the group of all permutations of A which fix the sets Fi×Oi for i∈{1,…,k} and which preserve the congruence ∼π.
Let N be the group of all permutations of A which fix all fibers of π setwise (i.e., N is the kernel of the map μπ:S→Aut⁡(𝔅)).

The following statements are direct consequences of the definitions above.

Proposition 4.14.

A first-order reduct C of A is a covering reduct of B with respect to the covering π if and only if Aut⁡(C)⊆S.

The group S can be written as a semidirect product N⋊Aut⁡(A).

Proof.

The first statement follows easily from the definition using that

Aut⁡(𝔅)=∏i=1kSym⁡(Oi).

Since N is the kernel of the homomorphism μπ:S→Aut⁡(𝔅), we have N◁S. It is obvious that Aut⁡(𝔄)≤S and that S=N⁢Aut⁡(𝔄). Since π is a trivial covering map, it follows that the kernel of μπ|Aut⁡(𝔄) is trivial, that is, N∩Aut⁡(𝔄)={idA}. ∎

The following lemma is a direct consequence of item (2) of Proposition 4.14. Let H and K be subgroups of the same group. Then we say that HnormalisesK if H is a subgroup of the normaliser of K, i.e., for every h∈H, we have that

{h-1⁢k⁢h∣k∈K}=K.

Lemma 4.15.

The mapping G↦G∩N defines a bijection between the closed subgroups of S that contain Aut⁡(A) and the closed subgroups of N which are normalised by Aut⁡(A). The inverse map is H↦H⋊Aut⁡(A).

Proof.

If H is a subgroup of N which is normalised by Aut⁡(𝔄), then the group generated by H and Aut⁡(𝔄) can be written as a product H⁢Aut⁡(𝔄). Since

H∩Aut⁡(𝔄)⊆N∩Aut⁡(𝔄)={idA},

it follows that this group can be written as a semidirect product H⋊Aut⁡(𝔄). Then (H⋊Aut⁡(𝔄))∩N=H.

We claim that if H is closed, then so is H⋊Aut⁡(𝔄). Let

α1,α2,…∈H⋊Aut⁡(𝔄)

be a sequence converging to some α∈Sym⁡(A). Let βi (and β) be the unique element in Aut⁡(𝔄) for which μπ⁢(βi)=μπ⁢(αi) (and μπ⁢(β)=μπ⁢(α)), that is, αi⁢βi-1∈H (and α⁢β-1∈H). Since μπ is continuous, it follows that (βi)i converges to β. Hence, the sequence (αi⁢βi-1)i converges to α⁢β-1. Since αi⁢βi-1∈H and H is closed, it follows that α⁢β-1∈H, and hence

α∈H⁢Aut⁡(𝔄)=H⋊Aut⁡(𝔄).

Therefore, H⋊Aut⁡(𝔄) is closed.

Let G be a subgroup of S containing Aut⁡(𝔄). We claim that Aut⁡(𝔄) normalises G∩N. Indeed, let g∈Aut⁡(𝔄). Then {g-1⁢h⁢g∣h∈G∩N}=G∩N since N is a normal subgroup of S which contains Aut⁡(𝔄). Since

(G∩N)∩Aut⁡(𝔄)⊆N∩Aut⁡(𝔄)={idA},

it follows that G can be written as G=(G∩N)⋊Aut⁡(𝔄). Moreover, it is clear that if G is closed, then so is G∩N.

We have obtained that the mappings defined in the lemma are inverses of each other, which also implies that they both are bijections. ∎

Hence, in order to classify the covering reducts of 𝔄, it is enough to classify those closed subgroups of N which are normalised by Aut⁡(𝔄). If K is a normal subgroup of a group G, then we write that two elements x1,x2∈Gare the same modulo K if they represent the same element in the factor group G/K, i.e., if x1⁢x2-1∈K.

Definition 4.16.

Let H be a subgroup of

∏i=1kSym⁡(Fi)⊆Sym⁡(F).

For i∈{1,…,k}, let Hi:=HF∖Fi|Fi⊆Sym(Fi), and let Ni be a subgroup of Hi normalised by H|Fi for every i∈{1,…,k} so that if |Oi|=1, we have Ni=Hi. We write N⁢(H,N1,…,Nk) for the group of all permutations α∈N such that

for every i∈{1,…,k} and for every xi∈Oi, there is a permutation γ∈H such that the action of α on the first coordinate of the fiber Fi×{xi} is exactly the i-th coordinate of γ,
for every i∈{1,…,k} and x,y∈Oi, the actions of α on the first coordinate of the fibers Fi×{x} and Fi×{y} are the same modulo Ni.

It follows directly from the definition that N⁢(H,N1,…,Nk) is a closed subgroup of N and normalised by Aut⁡(𝔄). We will show that subgroups of N with these properties are of the form N⁢(H,N1,…,Nk).

Definition 4.17.

Let G be a subgroup of N which is normalised by Aut⁡(𝔄).

Let H⁢(G) be the subgroup of ∏i=1kSym⁡(Fi) containing all permutations γ such that there exists a permutation α∈G and for every i∈{1,…,k} an element xi∈Oi such that the action of α on the first coordinate of the fiber Fi×{xi} is exactly the i-th coordinate of γ.
Hi(G):=H(G)F∖Fi|Fi.
Let Ni⁢(G) be the group of all γ∈Sym⁡(Fi) such that there exists a permutation α∈G and an x∈Oi such that the action of α on the first coordinate of the fiber Fi×{x} equals γ and α fixes every element of A∖(Fi×{x}).

Remark 4.18.

Since Aut⁡(𝔄)=∏i=1kSym⁡(Oi) normalises G, it does not matter which elements xi∈Oi we take in the definition of H⁢(G) and Ni⁢(G). It follows that H⁢(G), Hi⁢(G), and Ni⁢(G) are indeed groups.

Remark 4.19.

It is clear from the definition that Ni⁢(G)≤Hi⁢(G), and if |Oi|=1, then Ni⁢(G)=Hi⁢(G).

Example 4.20.

A simple example for k=1 where H1≠N1 is Example 2.21, where N1={id} and H1=ℤ2.

Lemma 4.21.

Let H,N1,…,Nk be the groups as introduced in Definition 4.16. Then H⁢(N⁢(H,N1,…,Nk))=H and Ni⁢(N⁢(H,N1,…,Nk))=Ni.

Proof.

It follows directly from the definition that H⁢(N⁢(H,N1,…,Nk))=H. For the second claim, let us assume that i∈{1,…,k}, x∈Oi, and that α∈N fixes every element of A∖(Fi×{x}). Let Hi be the group as introduced in Definition 4.16. We have to show that α∈N⁢(H,N1,…,Nk) if and only if α acts on the first coordinate of the fiber Fi×{x} as a permutation in Ni.

For the “if”, let us assume that the action of α on the first coordinate of the fiber Fi×{x}, which we denote by γi, is an element of Ni. We show that

α∈N⁢(H,N1,…,Nk).

First we show that α satisfies the first bullet of Definition 4.16. Arbitrarily choose j∈{1,…,k} and xj∈Oj. Let γj denote the action of α on the fiber Fj×{xj}, and let γ=∏j=1kγj. We have to show that γ∈H. It follows from the definition of α that γj=id⁡(Fj) if j≠i. Therefore, it is enough to show that

γi∈Hi=HF∖Fi|Fi.

If xi=x, then γi∈Ni, and if xi≠x, then by definition, γi=id⁡(Fi)∈Ni. In either case, we have γi∈Ni≤Hi. It remains to show that α satisfies the second bullet of Definition 4.16. In fact, we show something stronger. We show that, for every j∈{1,…,k} and y∈Oj, the action of α on Fj×{y} is an element of Nj. If y=x, and thus j=i, then this is exactly our assumption on α. If y≠x, then this action is trivial since, by definition, α fixes every element in A∖(Fi×{x}).

For the other direction, let us assume that α∈N⁢(H,N1,…,Nk). We distinguish two cases. If |Oi|=1, then by definition, α acts on the first coordinate of the fiber Fi×{x} as a permutation in Hi⁢(N⁢(H,N1,…,Nk)), and by Remark 4.19, we have

Ni⁢(N⁢(H,N1,…,Nk))=Hi⁢(N⁢(H,N1,…,Nk))=(H⁢(N⁢(H,N1,…,Nk))F∖Fi)|Fi=HF∖Fi|Fi=Hi=Ni,

where the last equality follows from the assumptions on the groups Hi and Ni in Definition 4.16.

If |Oi|>1, then let y be an element in Oi different from x. Then if

α∈N⁢(H,N1,…,Nk),

then on one hand, we know that the action of α on the first coordinate of the fibers Fi×{x} and Fi×{y} are the same modulo Ni. On the other hand, we know by the definition of α that its action on the fiber Fi×{y} is trivial. Therefore, the action of α on the first coordinate of the fiber Fi×{x} is an element of Ni. ∎

Definition 4.22.

Let Ni be a subgroup of Sym⁡(Fi) for some i∈{1,…,k}. Then N*⁢(N1,…,Nk) is defined to be the closure of the group generated by all permutations α for which there exists an i and x∈Oi such that the action of α on the first coordinate of the fiber Fi×{x} is in Ni and α fixes every element of A∖(Fi×{x}).

It follows easily from the definition that N*⁢(N1,…,Nk) is contained in every closed group G≤N normalised by Aut⁡(𝔄) with Ni⁢(G)=Ni. It is also easy to see that, in fact, N*⁢(N1,…,Nk)=N⁢(∏i=1kNi,N1,…,Nk) (using the notation from Definition 4.16).

Example 4.23.

Let 𝔅:=ℕ and 𝔄 a strongly trivial finite covering structure of 𝔅 with fibers of size four. Then the covering structure from Example 2.22 is a covering reduct ℭ of 𝔄. Let G:=Aut(ℭ). As G is transitive, we have S=N and k=1 in Definition 4.17. Then H⁢(G)=∏i=1kℤ4 and N1⁢(G)=ℤ2.

Lemma 4.24.

Let G≤N be normalised by Aut⁡(A). Then Ni⁢(G)◁H⁢(G)|Fi.

Proof.

Let α∈H⁢(G)|Fi and β∈Ni⁢(G). Let γ∈G be an element witnessing α∈H⁢(G)|Fi. Let x∈Oi, and let δ∈G be an element witnessing δ∈Ni⁢(G) on the fiber Fi×{x}. Then the element γ-1⁢δ⁢γ∈G witnesses the fact that

α-1⁢β⁢α∈Ni⁢(G).∎

Lemma 4.25.

Let G be a closed subgroup of N normalised by Aut⁡(A). Let α∈G and u,v∈Oi. Then the actions of α on the first coordinate of the fibers Fi×{u} and Fi×{v} are the same modulo Ni⁢(G).

Proof.

Let αu and αv denote the action of α on the first coordinate of the fibers of u and v, respectively, so αu,αv∈Sym⁡(Fi). For β∈Aut⁡(𝔅), we write π-1⁢(β) for the unique γ∈Aut⁡(𝔄) such that μπ⁢(γ)=β. Let β=(u⁢v)∈Aut⁡(𝔅). Let

γ:=α-1(π-1(β))-1απ-1(β).

Then the action of γ on the first coordinate of the fiber Fi×{u} is αu-1⁢αv. On the other hand, γ fixes every element of A∖Fi×{u,v}. Now let v1,v2,… be pairwise distinct elements of Oi. Let βi:=(uvi) and γi:=(π-1(βi))-1γπ-1(βi). Then γi acts on the first coordinate of the fiber Fi×{u} as αu-1⁢αv, and it fixes every element outside Fi×{u,vi}. Therefore, the permutations γi converge to a permutation γ′ which acts on the first coordinate of the fiber Fi×{u} as αu-1⁢αv, and fixes every element outside Fi×{u}. By our assumption, G is closed, so γ′∈G. By definition, this implies that αu-1⁢αv∈Ni⁢(G). ∎

Proposition 4.26.

Let C be a covering reduct of A and G:=Aut(C)∩N. Then

G=N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)).

Proof.

By Remark 4.19, we know that Ni⁢(G)≤Hi⁢(G)=H⁢(G)F∖Fi|Fi, and if |Oi|=1, then Ni⁢(G)=Hi⁢(G). Lemma 4.24 implies that H⁢(G)|Fi normalises Ni⁢(G). Therefore, the group G=N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)) is well-defined.

We first show that G≤N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)). Let α∈G. Then the definition of the group H⁢(G) implies that α satisfies the first item in the definition of N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)). By Lemma 4.25, α also satisfies the second item of this definition, and hence α∈N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)).

Now let α∈N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)) be arbitrary. Let ui∈Oi be arbitrary elements. By Lemma 4.21, we have

H⁢(N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)))=H⁢(G).

This implies that there exists an α′∈G such that, for every i∈{1,…,k}, the actions of α and α′ agree on Fi×{ui}. For v∈Oi, let αv and αv′ denote the action of α and α′, respectively, on the fiber Fi×{v}. We claim that, for all v∈Oi, it holds that αv⁢(αv′)-1∈Ni. By Lemma 4.21, we have

Ni⁢(N⁢(H⁢(G),N1⁢(G),…,Nk⁢(G)))=Ni⁢(G),

and hence, by Lemma 4.25, it follows that αv⁢αui-1∈Ni⁢(G), and

αv′⁢(αui′)-1=αv′⁢αui-1∈Ni⁢(G),

and hence

αv⁢(αv′)-1=αv⁢αui-1⁢αui⁢(αv′)-1=αv⁢αui-1⁢(αv′⁢αui-1)-1∈Ni⁢(G).

This implies that

α⁢(α′)-1∈N⁢(∏i=1kNi⁢(G),N1⁢(G),…,Nk⁢(G))=N*⁢(N1⁢(G),…,Nk⁢(G))⊆G.

Therefore, α=(α⁢(α′)-1)⁢α′∈G. ∎

Remark 4.27.

Proposition 4.26 generalises [67, Theorem 3.1] from Sym⁡(ℕ) to arbitrary automorphism groups of structures in 𝒰*.

Corollary 4.28.

𝔄 has finitely many covering reducts with respect to π.

Proof.

By Lemma 4.15 and item (1) of Proposition 4.14, it is enough to show that N has finitely many closed subgroups which are normalised by Aut⁡(𝔄). By Proposition 4.26 and Remark 4.19, these groups can be characterised by a subgroup H of ∏i=1kSym⁡(Fi) and a system of subgroups Ni≤Hi. Then the statement of the corollary follows from the fact that there are finitely many choices for these groups. ∎

5 Finite coverings of reducts of unary structures

In this section, we show that every structure in F⁢(R⁢(𝒰)) is a quasi-covering reduct (introduced in Definition 5.9) of a strongly trivial covering of some structure in 𝒰* (Proposition 5.11), and that there are only finitely many of such reducts for each structure in R⁢(𝒰) (Theorem 5.10). Moreover, we observe (Remark 5.8) that F⁢(R⁢(𝒰))⊆R⁢(F⁢(𝒰)).

5.1 The Ramsey property and canonical functions

Let 𝔄,𝔅 be structures. A function f:A→B is called canonical from 𝔄 to 𝔅 if, for every t∈An and α∈Aut⁡(𝔄), there exists β∈Aut⁡(𝔅) such that

f⁢(α⁢(t))=β⁢(f⁢(t)).

Hence, a canonical function f induces, for every n, a function from the orbits of n-tuples of Aut⁡(𝔄) to the orbits of n-tuples of Aut⁡(𝔅); these functions will be called the behaviour of f. Canonical functions as a tool to classify reducts of homogeneous structures with finite relational signature have been introduced in [17] and used in [59, 20, 1, 2, 13, 11]. The existence of certain canonical functions in the automorphism group of a structure 𝔄 is typically shown using Ramsey properties of 𝔄. We will not introduce Ramsey structures here; all that is needed is the well-known fact that (ℚ;<) is Ramsey, and the following result from [17]. A structure is called ordered if the signature contains a binary relation symbol that denotes a (total) linear ordering of the domain.

Lemma 5.1 (see [19]).

Let D be an ordered homogeneous Ramsey structure with finite relational signature, and let f:D→D be a function. Then there exists a function g∈{α∘f∘β∣α,β∈Aut⁡(D)}¯ which is canonical as a function from D to D.

The following is an easy consequence of the definitions.

Lemma 5.2.

Let A be a homogeneous structure with finite relational signature, and let B be a first-order reduct of A. If f and g are canonical functions from A to B with the same behaviour, then Aut⁡(B)∪{f}¯=Aut⁡(B)∪{g}¯.

The next lemma follows from the observation that if 𝔄 is homogeneous with a relational signature of maximal arity k, then the behaviour of a canonical function f from 𝔄 to 𝔅 is fully determined by the function induced by f on the orbits of k-tuples (see [22, in particular the comments at the end of Section 4.1]).

Lemma 5.3.

Let A be a homogeneous structure with finite relational signature, and let B be ω-categorical. Then there are finitely many behaviours of canonical functions from A to B.

The structures in F⁢(𝒰*) have homogeneous expansions with finite relational signature which we describe next.

Lemma 5.4.

Let B∈U*, and let π:A→B be a strongly trivial finite cover. Then

𝔄 is interdefinable with a homogeneous structure ℭ with finite relational signature,
𝔄 is a first-order reduct of an ordered homogeneous Ramsey structure 𝔇 with finite relational signature.

Proof.

Let O1,…,Ok be the orbits of 𝔅. Following Remark 4.11, we can assume that 𝔄=⊔i=1k(Fi×Oi) for some finite sets Fi, and that Aut⁡(𝔄) consists of all permutations which preserve the first coordinate and stabilise the sets Oi on the second coordinate. For each i≤k and s∈Fi, we define the unary relation Ui,s:={(s,u)∣u∈Oi}. Let ℭ be the relational structure with domain A and the relations Ui,s and ∼π. Then Aut⁡(ℭ)=Aut⁡(𝔄). Hence, 𝔄 and ℭ are interdefinable. It is easy to see that ℭ is homogeneous. This proves (1).

To prove item (2), we define an ordering < on 𝔅 as follows. For each infinite orbit Oi, let us fix an ordering <i on Oi which is isomorphic to (ℚ;<). Let us also fix an ordering of ≺i on Fi for all i. Then < is defined as follows.

If π⁢(x)∈Oi,π⁢(y)∈Oj and i<j, then x<y,
If π⁢(x),π⁢(y)∈Oi and π⁢(x)<iπ⁢(y), then x<y,
If π⁢(x)=π⁢(y)∈Oi and x′ and y′ are the projections of x and y to the first component, then x<y if and only if x′≺iy′.

To show that the expansion 𝔇 of ℭ by the ordering < has the Ramsey property, we use the fact that if a structure is the disjoint union of substructures induced by definable subsets, and the substructures are Ramsey, then the structure itself is Ramsey (see [10]). For each i≤k, let 𝔇i be the substructure of 𝔇 induced by π-1⁢(Oi). Note that π-1⁢(Oi)=⋃s∈FiUi,s and hence is definable in 𝔇. If Oi is infinite, then Aut⁡(𝔇i) is topologically isomorphic to Aut⁡(ℚ;<). The property of a structure of being Ramsey is a property of the automorphism group of the structure, viewed as a topological group (again, see [10]). It follows that 𝔇 is Ramsey. ∎

5.2 Reducts of finite covers of reducts of unary structures

Let 𝔅∈R⁢(𝒰), and let π:𝔄→𝔅 be a finite covering map. In this section, we study the closed supergroups G of Aut⁡(𝔄) that preserve ∼π such that μπ⁢(G) also preserves the congruence ∇⁡(𝔅).

Lemma 5.5.

Let B∈U*, and let π:A→B be a strongly trivial finite covering map. Let O1,…,Ok be the orbits of B, and let D be the ordered homogeneous finite signature Ramsey expansion of A from Lemma 5.4. Suppose that f∈Sym⁡(A) preserves ∼π and that μπ⁢(f) preserves the partition P:={O1,…,Ok}. Then the monoid M:=〈Aut⁡(A),f〉¯ contains a surjective map h which is canonical from D to A and such that μπ⁢(h) has the same action on P as μπ⁢(f).

Proof.

By Lemma 5.1, we obtain that there exists a function

g∈{α⁢f⁢β∣α,β∈Aut⁡(𝔇)}¯⊆M

which is canonical from 𝔇 to 𝔇. Since g∈M, it follows that g preserves ∼π. But note that g is not necessarily surjective. For m∈M, define μπ⁢(m) by

x↦π⁢(m⁢(π-1⁢(x)))

as in the case of bijective functions. Since every automorphism of 𝔇 preserves the orbits O1,…,Ok, it follows that if μπ⁢(f)⁢(Oi)=Oj, then μπ⁢(g)⁢(Oi)⊆Oj.

If μπ⁢(f)⁢(Oi)=Oj, then |FOi|=|FOj| since f preserves ∼π and is surjective. Therefore, for every g′∈M and u∈B, the restriction of g′ to U:=π-1(u) is a bijection between U and π-1⁢(μπ⁢(g′)⁢(u)). In particular, this holds for g∈M. If Oi={ui}, then g defines a bijection between π-1⁢(ui) and π-1⁢(μπ⁢(f)⁢(ui)). If Oi is infinite, then π-1⁢(μπ⁢(g)⁢(Oi)) is a union of infinitely many classes of ∼π. Moreover, Oi is infinite if and only if μπ⁢(g)⁢(Oi) is infinite. Let e:A→A be defined as (s,u)↦(s,μπ⁢(g)⁢(u)). Let ℭ be the homogeneous structure from Lemma 5.4 which has the property that Aut⁡(ℭ)=Aut⁡(𝔄). Then e is an isomorphism between ℭ and the substructure of ℭ induced by g⁢(C). Since ℭ is homogeneous, it follows that e∈Aut⁡(ℭ)¯=Aut⁡(𝔄)¯, and so there is a sequence e1,e2,…∈Aut⁡(𝔄) which converges to e. Then hi:=ei-1g∈M converges to h:=e-1∘g, and thus h∈M. We claim that the mapping h satisfies the conditions of the lemma. By definition, h⁢(A)=e-1⁢(g⁢(A))=A, that is, h is surjective. Since e-1 preserves the relations of ℭ, it follows that the mapping h is canonical from 𝔇 to ℭ (and therefore also from 𝔇 to 𝔄). For the same reason, μπ⁢(e-1) preserves all orbits Oi. This implies that μπ⁢(h) and μπ⁢(g) and therefore also μπ⁢(f) have the same action on {O1,…,Ok}. ∎

Lemma 5.6.

Let B∈U*, and let π:A→B be a finite covering map. Moreover, let O1,…,Ok be the orbits of B. Then Sym⁡(A) has finitely many closed subgroups G such that

Aut⁡(𝔄)⊆G,
G preserves ∼π,
μπ⁢(G) preserves the partition {O1,…,Ok} of B.

Proof.

By Proposition 4.9, we know that 𝔄 is a covering reduct of some strongly trivial covering ℭ of 𝔅 (with respect to π). Then let 𝔇 be the ordered homogeneous finite-signature Ramsey expansion of ℭ from Lemma 5.4. Let G be a closed subgroup of Sym⁡(A) as in the formulation of the lemma. Then G acts on the set {O1,…,Ok}. Let K be the kernel of this action. Then K is closed, and the index of K in G is finite. Also, K⊆S, where S is the group as in Definition 4.13. Therefore, K is the automorphism group of a covering reduct of 𝔄 (Proposition 4.14). Then, by Theorem 4.2, there are finitely many possible choices for the group K. By Lemma 5.5, for each f∈G, there exists a surjective map h∈〈K,f〉¯ which is canonical from 𝔇 to 𝔄 such that f and h induce the same permutation σ of {O1,…,Ok}. We claim that K∪{f} and K∪{h} generate the same group. The image of the action of K∪{f} and K∪{h} on {O1,…,Ok} is 〈σ〉, and the kernel of these actions is again K. Therefore,

(5.1)[〈K∪{f}〉:K]=[〈K∪{h}〉:K]=l,

where l is the order of the permutation σ. In particular, the groups 〈K∪{f}〉 and 〈K∪{h}〉 are closed. Hence, h∈K∪{f}, and thus 〈K∪{f}〉≤〈K∪{h}〉. Then, by using equality (5.1) again, it follows that 〈K∪{f}〉=〈K∪{h}〉.

Since [G:K] is finite, each group G is generated by finitely many (at most k!) elements over K. By the previous paragraph, we can assume that each of these generators are canonical from 𝔇 to 𝔄. There are finitely many possible behaviours of canonical functions from 𝔇 to 𝔄 (Lemma 5.3). If two functions have the same behaviour, they generate the same group over Aut⁡(𝔄) (Lemma 5.2). This implies that there are finitely many choices for the group G. ∎

Theorem 5.7.

Let B∈R⁢(U), and let π:A→B be a finite covering map. Then Aut⁡(A) has finitely many closed supergroups G such that G preserves ∼π and μπ⁢(G) preserves ∇⁡(B).

Proof.

Let O1,…,Ok be the classes of ∇⁡(𝔅). Then, by Lemma 3.3, it follows that ∏i=1kSym⁡(Oi)⊆Aut⁡(𝔅). Let 𝔅′ be a structure with

Aut⁡(𝔅′)=∏i=1kSym⁡(Oi).

Then 𝔅′∈𝒰* by Lemma 2.14 and ∇⁡(𝔅)=∇⁡(𝔅′). The group Aut⁡(𝔄) acts naturally on the set {O1,…,Ok}. Let K be the kernel of this action, and let 𝔄′ be a structure so that Aut⁡(𝔄′)=K. The action of Aut⁡(𝔄′) on B equals Aut⁡(𝔅′). Therefore, π:𝔄′→𝔅′ is a finite cover. Then the statement of the theorem follows from Lemma 5.6 and from the fact that the orbits of 𝔅′ are exactly the classes of the congruence ∇⁡(𝔅). ∎

Proposition 5.8.

F⁢(R⁢(𝒰))⊆R<∞⁢(F⁢(𝒰*)).

Proof.

Let 𝔄∈F⁢(R⁢(𝒰)). Following the notation of the proof of Theorem 5.7, we have that [Aut(𝔄):Aut(𝔄′)]=[Aut(𝔄):K] is finite since K is defined as the kernel of the action of Aut⁡(𝔄) on the set {O1,…,Ok}. As we saw in the proof of Theorem 5.7, we have 𝔄′∈F⁢(𝒰*). Hence, 𝔄∈R<∞⁢(F⁢(𝒰*)). ∎

Later, we will see (in Theorem 6.29) that, in fact, F⁢(R⁢(𝒰))=R<∞⁢(F⁢(𝒰*)). The following definition of quasi-covering reducts is needed for a model-theoretic reformulation of Theorem 5.7, which is given in Theorem 5.10 below.

Definition 5.9.

Let 𝔅 be ω-categorical, and let π:𝔄→𝔅 be a finite cover. A first-order reduct ℭ of 𝔄 is called a quasi-covering reduct of 𝔄 with respect to π if Aut⁡(ℭ) preserves ∼π and μπ⁢(Aut⁡(ℭ))⊆Sym⁡(B) preserves ∇⁡(𝔅).

Theorem 5.10.

Let B∈R⁢(U), and let π:A→B be a finite cover. Then A has finitely many quasi-covering reducts with respect to π.

Proof.

The statement follows immediately from Theorem 5.7 and Definition 5.9. ∎

Proposition 5.11.

Let B∈R⁢(U), and let π:A→B be a finite cover. Then A is a quasi-covering reduct of a strongly trivial covering of some structure in U*.

Proof.

Let us define the structures 𝔄′ and 𝔅′ as in the proof of Theorem 5.7. Then 𝔄 is a finite quasi-covering reduct of 𝔄′ with respect to the covering map π. The map π:𝔄′→𝔅′ is a finite covering, and 𝔅′∈𝒰*. By Proposition 4.9, 𝔄′ is a covering reduct of some strongly trivial covering of 𝔅′. Therefore, 𝔄 is a quasi-covering reduct of a strongly trivial covering of 𝔅′∈𝒰*. ∎

6 Structures with small orbit growth

In this section, we show that 𝒦exp+=(F∘R)⁢(𝒰). We start in Section 6.1 with some observations from enumerative combinatorics that we need to obtain information about ∇⁡(𝔄) if 𝔄∈𝒦exp+. In Section 6.2, we discuss the effect of stabilising a group from 𝒢exp+ at finitely many constants. Section 6.3 treats the case that 𝔄 is primitive; here we rely on work of Macpherson [55]. We then focus on permutation groups G in 𝒢exp+ where the congruence Δ⁢(G) is trivial, i.e., Δ⁢(G) is the identity congruence (Section 6.4); the general case is treated in Section 6.5. In Section 6.6, we use these results to prove Thomas’ conjecture for all structures in the class 𝒦exp+.

6.1 Growth rates for partitions

For n,k∈ℕ, let pk⁢(n) be the number of partitions of the set {1,…,n} with parts of size at most k; this is the Sloane integer sequence A229223. Asymptotic formulas for pk⁢(n) are known for k∈{1,…,4} (called allied Bell numbers in a letter of John Riordan). We need an upper and a lower bound for all k∈ℕ.

Lemma 6.1.

Let ε>0. Then pk⁢(n)≥n(k-1k-ε)⁢n if n is large enough.

Proof.

Let sk⁢(n) be the number of partitions of {1,…,k⁢n} where all the parts contain exactly k elements. Clearly, sk⁢(1)=1 for all k∈ℕ. To form a partition of {1,…,k⁢n} for n>1, we first choose the class containing the number kn, and then we choose a partition of the remaining elements. Hence, sk satisfies the recursion

sk⁢(n)=(k⁢n-1k-1)⁢sk⁢(n-1).

Since (k⁢n-1k-1)≥nk-1, we obtain by induction that

sk⁢(n)≥nk-1⁢(n-1)k-1⁢⋯⁢2k-1=(n!)k-1.

Stirling’s formula (n!∼2⁢π⁢n⁢(ne)n for n tending to infinity) implies that

(n!)k-1≥n(k-1)⁢(1-ε′)⁢n

for any ε′>0 if n is large enough. Hence,

pk⁢(n)≥sk⁢(⌊nk⌋)≥⌊nk⌋(k-1)⁢(1-ε′)⁢⌊nk⌋≥(nk-1)(k-1)⁢(1-ε′)⁢(nk-1)≥n(k-1)⁢(1-ε′)⁢(1-ε′′)⁢1k⁢n≥n(k-1k-ε)⁢n

for an appropriate choices of ε′,ε′′>0 if n is large enough. ∎

Lemma 6.2.

Let n,k∈N. If d>k-1k, then pk⁢(n)<c⁢nd⁢n for some c.

Proof.

To form a partition of {1,…,n} for n>1, we first choose the class containing the number n, and then we choose a partition of the remaining elements. We thus have the following recursion formula:

(6.1)pk⁢(n)=∑i=0k-1(n-1i)⁢pk⁢(n-1-i).

We claim that the following inequality holds if n is large enough:

(6.2)∑i=0k-1(n-1i)⁢(n-1-i)d⁢(n-1-i)<nd⁢n.

In order to prove this, it is enough to show that if i≤k-1 and n is large enough, then

(n-1i)⁢(n-1-i)d⁢(n-1-i)<1k⁢nd⁢n,

that is,

(6.3)(n-1i)<1k⁢(nn(n-1-i)n-1-i)d.

We have

nn(n-1-i)n-1-i=∏j=0i(n-j)n-j(n-j-1)n-j-1=∏j=0i((n-j)⁢(n-jn-j-1)n-j-1)≥∏j=0i(n-j)=n⁢(n-1)⁢⋯⁢(n-i).

This implies that, in order to show inequality (6.3), it is enough to show that

(n-1i)<1k⁢(n⁢(n-1)⁢⋯⁢(n-i))d

if n is large enough. By rearranging the inequality above, we obtain that it is equivalent to the following inequality:

(6.4)1i!⁢((n-1)⁢⋯⁢(n-i))1-d<1k⁢nd.

The LHS of inequality (6.4) is asymptotically 1i!⁢ni⁢(1-d). By our assumption, we have d>k-1k; thus i⁢(1-d)<ik≤k-1k<d. This implies inequality (6.4) and hence inequality (6.2) if n is large enough.

Now let us choose an N so that inequality (6.2) holds for all n>N, and then let us choose a c so that pk⁢(n)<c⁢nd⁢n holds for n≤N. Then we show that pk⁢(n)<c⁢nd⁢n also holds for n>N by induction on n. Suppose that we already know that pk⁢(m)<c⁢md⁢m holds for all m<n. Then, by using the recursion formula (6.1) and inequality (6.2), we obtain

pk⁢(n)=∑i=0k-1(n-1i)⁢pk⁢(n-1-i)<c⁢∑i=0k-1(n-1i)⁢(n-1-i)d⁢(n-1-i)<c⁢nd⁢n.∎

6.2 The number of ∇-classes in point stabilisers

In this section, we examine the possible growth of the number of ∇-classes in stabilisers of finite sets.

Lemma 6.3.

Let G∈Gexp+ be a permutation group on a countably infinite set X, that is, oni⁢(G)≤c1⁢nd⁢n for some c1,d with d<1. Let F⊂X be finite. Then, for every ε>0,

there exists a constant c2 such that oni⁢(GF|X∖F)<c2⁢n(d+ε)⁢n,
there exists a constant c3 such that oni⁢(GF)<c3⁢n(d+ε)⁢n.

In particular, GF∈Gexp+ and GF|X∖F∈Gexp+.

Proof.

Let ε>0. The orbits of injective n-tuples of GF can be embedded into the orbits of injective (n+|F|)-tuples of G by mapping the orbit of a tuple t into the orbit of (t,t′), where t′ is any |F|-tuple such that (t,t′) has pairwise distinct entries and all elements of F appear in (t,t′). Hence,

oni⁢(GF)≤on+|F|i⁢(G)≤c1⁢(n+|F|)d⁢(n+|F|)≤c2⁢n(d+ε)⁢n

for an appropriate constant c2. Choosing ε>0 such that d+ε<1 shows that GF∈𝒢exp+. The statements for GF|X∖F can be shown analogously. ∎

Definition 6.4.

Let G be an oligomorphic permutation group on a countably infinite set X. For every finite set F⊂X, let mG⁢(F) be the number of ∇⁡(GF)-classes. For n∈ℕ, let mG(n):=max({mG(F)∣F⊂X,|F|=n}).

Remark 6.5.

If F1,F2⊂X are contained in the same orbit of n-subsets of G, then mG⁢(F1)=mG⁢(F2). Hence, the set {mG⁢(F)∣F⊂X,|F|=n} is finite, and so the maximum of this set always exists.

Lemma 6.6.

Let G be a permutation group. Suppose that oni⁢(G)≤c1⁢nd⁢n for some c1 and d<1. Then, for every ε>0, we have mG⁢(n)≤c2⁢nd+ε for some constant c2.

Proof.

Suppose that G is a permutation group on X, and let F⊂X be of size n. Suppose that X1,X2,…,Xl are the infinite classes of the congruence ∇⁡(GF), and arbitrarily choose xi∈Xi for i∈{1,…,l}. Then, for each function

f:{1,…,k}→{1,…,l},

there are pairwise distinct elements y1,…,yk so that yj∈Xf⁢(j)∖{xf⁢(j)}. Let tf:=(x1,…,xl,y1,…,yk). Then the tuples tf are injective and lie in pairwise different orbits of GF|X∖F. Thus, ln≤oni⁢(GF|X∖F)≤c2⁢n(d+ε)⁢n for some c2 by Lemma 6.3. Thus, l≤c2⁢nd+ε, and therefore mG⁢(F)≤c2⁢nd+ε. ∎

Lemma 6.7.

Let G be a permutation group on a countably infinite set X, and suppose that oni⁢(G)≤c⁢nd⁢n for some c and d<1. Let F⊂X be finite, let R be a congruence of GF, and let k-1k>d. Then R has finitely many classes of size at least k.

Proof.

By Lemma 6.3, we can assume that F=∅. Suppose to the contrary that R has infinitely many classes of size at least k. Let n be arbitrary, and let 𝒫nk be the set of partitions P={S1,…,Sl} of {1,2,…,n} such that |Si|≤k for all i=1,…,l. For each P∈𝒫nk, choose pairwise distinct elements x1P,…,xnP∈X such that xiP⁢R⁢xjP if and only if xiP and xjP are in the same set in the partition P. Then the n-tuples (x1P,…,xnP) for P∈𝒫nk are injective and lie in pairwise different orbits of G. Therefore, oni⁢(G)≥|𝒫nk|. Let us choose ε>0 such that k-1k-ε>d. Then, by Lemma 6.1, it follows that

oni⁢(G)≥|𝒫nk|=pk⁢(n)≥n(k-1k-ε)⁢n>c⁢nd⁢n

for n large enough. This contradicts our assumption. ∎

Corollary 6.8.

Let G be a permutation group on a countably infinite set X, and suppose that oni⁢(G)≤c⁢nd⁢n for some c and d<1. Let F⊂X be finite, and let R be a congruence of GF. Then R has finitely many infinite classes.

Proof.

Follows directly from Lemma 6.7. ∎

Definition 6.9.

Let (𝒢exp+)k, k∈ℕ, denote the class of those groups G∈𝒢exp+ for which the following holds.

($*^{k}$)For every finite⁢F⊂X,every congruence of⁢GF⁢has at mostfinitely many equivalence classes of size at least⁢k.

Let (𝒦exp+)k denote the classes of those structures in 𝒦exp+ whose automorphism group is in (𝒢exp+)k.

Using the definition above, Lemma 6.7 immediately implies the following.

Corollary 6.10.

𝒢exp+=⋃k=1∞(𝒢exp+)k, and Kexp+=⋃k=1∞(Kexp+)k.

6.3 The primitive case

We use the following theorem of Dugald Macpherson [55].

Theorem 6.11 ([55, Theorem 1.2]).

Let G be a permutation group on a countably infinite set X which is primitive but not highly transitive. Then there is a polynomial p such that oni⁢(G)≥n!p⁢(n).

Theorem 6.11 immediately implies the following.

Lemma 6.12.

Let G∈Gexp+ be primitive. Then G is highly transitive.

Proof.

Stirling’s formula implies that, for all c and d<1 and every polynomial p, if n is large enough, then n!p⁢(n)>c⁢nd⁢n. Hence, the lemma follows from Theorem 6.11. ∎

6.4 The case when Δ⁢(G) is trivial

The result of Macpherson (Theorem 6.11) is used via the following lemma.

Lemma 6.13.

Let G∈Gexp+ be such that Δ⁢(G) is trivial and such that G stabilises each class of ∇⁡(G) setwise. Then G acts highly transitively on each of its orbits.

Proof.

Let O1,…,Om be the orbits of G. Then O1,…,Om are also the classes of ∇⁡(G). We claim that the action of G on Oi is primitive for each i∈{1,…,m}; this suffices, because then the statement of the lemma follows from Lemma 6.12.

Let Ri be a congruence of G|Oi. Since G acts transitively on Oi, it follows that every class of Ri has the same size. If this size is finite, then let us consider the congruence Ri*:=Ri∪{(x,x)∣x∈X∖Oi}. Then every class of Ri* is finite and thus Ri* must be finer than Δ⁢(G). Since Δ⁢(G) is the identity congruence, Ri* and Ri are the identity congruence, too.

Now assume that every class of Ri is infinite. Then, by Corollary 6.8, Ri has finitely many classes. Let C1,C2,…,Cl be these classes. Then

{O1,…,Oi-1,Oi+1,…,Om,C1,C2,…,Cl}

is an invariant partition. Since ∇⁡(G) is the finest congruence with finitely many classes, it follows that l=1, and thus Ri is again the identity congruence. Therefore, G|Oi is primitive for all i. ∎

Under the conditions of Lemma 6.13, we will show that, in fact, if G is closed, then G=∏i=1mSym⁡(Oi), where O1,…,Om are the orbits of G, that is, G is the automorphism group of a unary structure (Lemma 6.22). The following lemma is well known (see, e.g., [58, Proposition 1.4 (2)] or [29, Corollary 2.2]).

Lemma 6.14.

Every normal subgroup of a highly transitive permutation group acting on an infinite set is either highly transitive or trivial.

Lemma 6.15.

Let G be a closed permutation group on a countably infinite set X. Let T be an infinite orbit of G such that G|T is highly transitive, and S:=X∖T. Then one of the following holds.

{idS}×Sym⁡(T)⊆G.
There exists a surjective homomorphism e:G|S→G|T such that a permutation γ of X is in G if and only if there exists a permutation α∈G|S so that γ|S=α and γ|T=e⁢(α).

Proof.

If α∈Sym⁡(S) and β∈Sym⁡(T), then we use the notation (α,β) for the unique permutation γ∈Sym⁡(X) whose restriction to S equals α and whose restriction to T equals β.

Case 1. For every α∈G|S, there is a unique e⁢(α)∈G|T such that (α,e⁢(α))∈G.

It is easy to see that, in this case, e is a surjective homomorphism from G|S to G|T; therefore condition (2) holds.

Case 2. For some α∈G|S, there exist at least two distinct permutations β1,β2∈G|T such that γ1:=(α,β1)∈G and γ2:=(α,β2)∈G.

Let K:={β∈Sym(T)∣(idS,β)∈G}. Then

K is nontrivial since β1⁢β2-1∈K,
K is closed in Sym⁡(T),
K is a normal subgroup of Sym⁡(T).

To prove normality, let (id,β)∈G, and let δ∈Sym⁡(T) be arbitrary. Since G|T is dense in Sym⁡(T), there is a sequence δ1,δ2,… of elements of G|T which converges to δ∈Sym⁡(T). By the definition of G|T, we know that there exist elements αi∈Sym⁡(S) such that ηi:=(αi,δi)∈G for every i∈ℕ. Then

G∋ηi⁢(id,β)⁢ηi-1=(id,δi⁢β⁢δi-1).

Therefore, limi⁡(id,δi⁢β⁢δi-1)=(id,δ⁢β⁢δ-1)∈G since G is closed. By definition, this implies that δ⁢β⁢δ-1∈K which shows that K is indeed a normal subgroup. By Lemma 6.14, K=Sym⁡(T). Thus, {idS}×Sym⁡(T)⊆G, i.e., condition (1) holds. ∎

Lemma 6.16.

Let G be a closed oligomorphic permutation group on a countably infinite set X. Let O1,…,Om be the orbits of G. Let us suppose that each Oi is infinite and that G acts highly transitively on each Oi. Let l∈{1,…,m}, and let S:=⋃i=1lOi be such that aclG⁡(S)=X. Then H:=G|S is closed in Sym⁡(S).

Proof.

We first show the statement for l=m-1. Let T:=Om (so that we have the same notation as in Lemma 6.15). First, let us assume that Lemma 6.15 (1) holds. Let (αj)j∈ℕ be a sequence that converges in G|S to α∈Sym⁡(S). Let βj∈{idS}×Sym⁡(T)⊆G be such that αj|T=βj|T, and let αj′:=αjβj-1∈G. Then αj′→(α,id⁡(T))∈G since G is closed. In particular, α∈G|S and H is closed.

Otherwise, if condition (1) of Lemma 6.15 does not hold, then by Lemma 6.15, we can assume that item (2) of Lemma 6.15 holds. Let e:G|S→G|T be as in item (2) of Lemma 6.15. If F⊂S is finite and α∈G|S, then

acl⁡(α⁢(F))∩T=acl⁡((α,e⁢(α))⁢(F))∩T=(α,e⁢(α))⁢(acl⁡(F))∩T=(α,e⁢(α))⁢(acl⁡(F)∩T)=e⁢(α)⁢(acl⁡(F)∩T).

By assumption, acl⁡(F)∩T is non-empty for some F (and it is always finite). Let

k:=|acl(F)∩T|.

Then, by our previous observation and the fact that G|T=e⁢(G|S) is highly transitive, it follows that, for any subset F′ of T of size k, there exists a finite subset F′′ of S such that acl⁡(F′′)∩T=F′.

We claim that, for all x∈T, there is a finite set F of S such that, for all α∈(G|S)F, it holds that e⁢(α)⁢(x)=x. Let F1′ and F2′ be subsets of T of size k such that F1′∩F2′={x}. As we have seen, there exist finite subset F1′′ and F2′′ of S such that acl⁡(Fi′′)∩T=Fi′ for i=1 and i=2. Now let F:=F1′′∪F2′′. Then if α∈(G|S)F, then α∈(G|S)Fi′′, so e⁢(α)⁢(Fi′)=Fi′. Therefore, we have e⁢(α)⁢(x)∈F1′∩F2′={x}, that is e⁢(α)⁢(x)=x.

Now let (αj)j be a convergent sequence in G|S. We want to show that the sequence (e⁢(αj))j is also convergent, i.e., for all x∈T, we have

e⁢(αj)⁢(x)=e⁢(αj+1)⁢(x)

if j is large enough. By our claim, it follows that there is a finite set F⊂S such that, for all α∈(G|S)F, we have e⁢(α)⁢(x)=x. Since (αj)j is a convergent, there is an index l such that αj⁢(y)=αj+1⁢(y) for all y∈F and l≤j. Then if j≥l, it follows that αj⁢αj+1-1∈H|F; hence e⁢(αj)⁢(e⁢(αj+1))-1⁢(x)=e⁢(αj⁢αj+1-1)⁢(x)=x, and thus e⁢(αj)⁢(x)=e⁢(αj+1)⁢(x). Therefore, (e⁢(αj))j is convergent, which shows that G|S is closed.

For l<m-1, note that S⊆P:=O1∪⋯∪Om-1. Hence, acl⁡(P)=X, and we can apply the above argument for P instead of S. We obtain that G|P is closed. Hence, the group G|P satisfies all the assumptions for G but has fewer orbits, so by induction, we finally obtain that G|S is closed. ∎

In the proof of the next lemma, it will be convenient to use a recent general result of Paolini and Shelah. A closed subgroup G of Sym⁡(X) has the

small index property if every subgroup of G of index less than 2ℵ0 is open, i.e., contains the pointwise stabiliser of a finite set F⊂X,
strong small index property if every subgroup of G of index less than 2ℵ0 lies between the pointwise and the setwise stabiliser of a finite set F⊂X.

The strong small index property of Sym⁡(X) itself has been shown in [37]. (In fact, all automorphisms of ω-categorical ω-stable structures, and thus, by the results that we are about to prove, all groups in 𝒢exp+, have the small index property [46].) On the other hand, already R⁢(𝒰) contains structures whose automorphism groups do not have the strong small index property (take, e.g., an equivalence relation with two infinite classes). A permutation group G on a set X is said to have no algebraicity if aclG⁡(Y)=Y for every Y⊆X. The following has been proved in [60, Corollary 2].

Theorem 6.17 ([60]).

Let X1 and X2 be countable, and let G1≤Sym⁡(X1) and G2≤Sym⁡(X2) be closed oligomorphic groups which have the strong small index property and have no algebraicity. Let us suppose that ξ is a topological isomorphism from G1 to G2. Then there exists a bijection b from X1 to X2 that induces ξ, i.e., for all x∈X2 and α∈G1, we have

(ξ⁢α)⁢(x)=b⁢(α⁢(b-1⁢(x))).

The small index property for G implies that, for a countable set Y, every homomorphism h:G→Sym⁡(Y) is continuous (this is easy to see and well known; see, e.g., [62, 57]). It follows from [43] that the image of a continuous homomorphism from Sym⁡(X) to Sym⁡(Y) is closed in Sym⁡(Y) (see [7, Theorem 1.3] for a much more general recent result which also implies this).

Lemma 6.18.

Let G be a closed oligomorphic permutation group on a countably infinite set X. Let O1,…,Om be the orbits of G. Let us suppose that each Oi is infinite, G acts highly transitively on each Oi, and for some l<m and S:=⋃i=1lOi, we have aclG⁡(S)=X and G|S=Sym⁡(O1)×…×Sym⁡(Ol). Then Δ⁢(G) is not trivial.

Proof.

Let j∈{l+1,…,m} and T:=Oj. Let Gj:=G|S∪T. Then Gj is closed by Lemma 6.16, and we can apply Lemma 6.15 to the group Gj with respect to the partition of S∪T into S and T. Since T⊆aclG⁡(S), it follows that condition (1) of Lemma 6.15 cannot hold. Thus, by Lemma 6.15, there exists a homomorphism ej:G|S→Sym⁡(Oj) so that Gj={(α,ej⁢(α))∣α∈G|S}. For i∈{1,…,l} and α∈G|Oi=Sym⁡(Oi), let α^i denote the unique permutation of S for which α^i|Oi=α and α^i|Ok=idOk if k≠i. Then define the homomorphisms

ei⁢j:Sym⁡(Oi)→Sym⁡(Oj),α↦ej⁢(α^i).

As mentioned before the lemma, the map ei⁢j is continuous. Let

Hi:={α^i∣α∈Sym(Oi)}.

Then Hi◁G|S, and so ej⁢(Hi)◁ej⁢(G|S). By definition, it follows that

ej⁢(G|S)=(Gj)|Oj=G|Oj.

In particular, ej⁢(G|S)≤Sym⁡(Oj) is highly transitive. Thus, by Lemma 6.14, it follows that either ej⁢(Hi) is also highly transitive or it is trivial. If ei⁢j is trivial for every i∈{1,…,l}, then G fixes every element of Oj contradicting the fact that G acts transitively on T=Oj. Thus, there is an i∈{1,…,l} such that the image I≤Sym⁡(Oj) of ei⁢j is highly transitive. As we have mentioned before the statement of the lemma, I is a closed subset of Sym⁡(Oj), so we can apply Theorem 6.17 and obtain a bijection bj between Oi and Oj which induces ei⁢j (we could as well have derived this from the argument in [45, Example 2, page 224]). Now let i′≠i, and let α be a nontrivial permutation of Oi′. Then α^i′ commutes with every element of Hi, and so ei′⁢j⁢(α) commutes with every element of ej⁢(Hi)=Sym⁡(Oj). Therefore, ei′⁢j⁢(α)=idOj. We have obtained that, for every j∈{l+1,…,m}, there is a unique i⁢(j)∈{1,…,l} and a bijection bj:Oi⁢(j)→Oj such that, for all g∈G and x∈Oj, we have g⁢(x)=bj⁢(g⁢(bj-1⁢(x))). In fact, the same is true in the case when j≤l: we can choose i⁢(j) to be j, and bj to be the identity map. Let b be the union of the functions b1,…,bm, and define the relation ∼ by x∼y⇔b⁢(x)=b⁢(y). Then ∼ is a congruence of G all of whose classes are finite. Moreover, ∼ is nontrivial since m>l. This implies that also Δ⁢(G) is nontrivial. ∎

Lemma 6.19.

Let H be an oligomorphic permutation group on a countably infinite set X with two infinite orbits Y and Z. Let us assume that H acts 2-transitively on Z and that there exists y∈Y so that |∇⁡((Hy)|Z)|≥2. Then, for every n∈N, there exist y1,…,yn∈Y such that |∇⁡((Hy1,…,yn)|Z)|≥n+1.

Proof.

By transitivity, we know that, for all y′∈Y, it holds that |∇⁡(Hy′|Z)|≥2.

We show the statement of the lemma by induction on n. For n=1, the statement is trivial. Now suppose that we know there exist y1,…,yn-1∈Y such that ∇⁡((Hy1,…,yn-1)|Z) has at least n classes. Let C1,…,Cm be the classes of ∇⁡((Hy1,…,yn-1)|Z). Let z1,z2∈C1. Since H acts 2-transitively on Z, it follows that there exists a yn∈Y such that (z1,z2)∉∇⁡(Hyn|Z). Let

R:=∇((Hy1,…,yn-1)|Z)∩∇(Hyn|Z).

Then R is a congruence of Hy1,…,yn which is strictly finer than (Hy1,…,yn-1,yn) and has finitely many classes. Therefore, we have that ∇⁡((Hy1,…,yn)|Z) is also finer than ∇⁡((Hy1,…,yn-1)|Z). In particular,

|∇⁡((Hy1,…,yn)|Z)|≥m+1≥n+1.∎

Lemma 6.20.

Let G∈Gexp+ be closed. Suppose that Δ⁢(G) is trivial and that G stabilises each class of ∇⁡(G) setwise. Let O1,…,Ol be the orbits of G. Suppose that each Oi is infinite and that S:=X∖Ol is algebraically closed. Then {idS}×Sym⁡(Ol)⊆G.

Proof.

By our assumptions, the orbits O1,…,Ol are the classes of ∇⁡(G). By Lemma 6.13, it follows that G acts highly transitively on each orbit Oi. We apply Lemma 6.15 to S and T:=Ol. If item (1) of Lemma 6.15 applies, then we are done. We claim that item (2) of Lemma 6.15 cannot hold. For this, it suffices to show that G contains a nontrivial permutation γ such that γ|S=idS. Indeed, if there exists a homomorphism e:G|S→G|T as in item (2) of Lemma 6.15, then (using the notation from the proof of Lemma 6.15) γ=(idS,e⁢(idS))=(idS,idT) is trivial.

Claim 1.

For every finite F⊆S and L:=GF|T, the congruence ∇⁡(L/Δ⁢(L)) is universal.

Proof of Claim 1.

Suppose to the contrary that there exists a finite F⊆S such that ∇⁡(L/Δ⁢(L)) is not universal; choose a minimal F with this property. For some y∈F, put F′:=F∖{y} and

E:=GF′|T,Z:=T/Δ(E),and K:=E/Δ(E)≤Sym(Z).

Let us consider the mapping π:T→Z which maps each element to its Δ⁢(E)-class. Then if u,v∈Zn are in different orbits of K, then all tuples in π-1⁢(u) are in different orbits of G than the tuples from π-1⁢(v). Moreover, if u∈Zn is injective, then so are all the tuples in π-1⁢(u). This means that the number of injective n-orbits of K is at most oni⁢(GF′). By Corollary 6.3, it follows that oni⁢(GF′)≤c⁢nd⁢n for some constants c,d with d<1. Therefore, oni⁢(K)≤c⁢nd⁢n, and thus K∈𝒢exp+. From the definition of K, it is clear that Δ⁢(K) is the identity congruence. It follows from the minimal choice of F that the congruence ∇⁡(K) is the universal congruence. Therefore, Lemma 6.13 implies that K is highly transitive.

Now let Y:=Oj∖F′, where j is such that y∈Oj. Then GF′ acts naturally on Y⊔Z. Let H be the image of this action (as a subgroup of Sym⁡(Y⊔Z)); note that H is highly transitive on Z. We claim that the group H, the orbits Y,Z and the element y∈Y satisfy the conditions of Lemma 6.19. The only nontrivial fact that we have to check is that the congruence ∇⁡(Hy|Z) is not universal. We know that ∇⁡(L/Δ⁢(L)) is not universal. By Lemma 2.15, this is equivalent to the fact that the congruence generated by ∇⁡(L) and Δ⁢(L) is not universal. The congruence Δ⁢(E) is also a congruence of L with finite classes; hence Δ⁢(E) is finer than Δ⁢(L). Hence, the congruence generated by ∇⁡(L) and Δ⁢(E) is finer than the congruence generated by ∇⁡(L) and Δ⁢(L), and thus it is also not universal. Using Lemma 2.15 again, it follows that ∇⁡(L/Δ⁢(E)) is not universal. Since L=(GF′)y|T, this means that the congruence ∇⁡(Hy|Z) is not universal.

If we apply Lemma 6.19, we obtain that, for every n, there exist y1,…,yn∈Y so that |∇⁡(Hy1,…,yn|Z)|≥n+1. This also implies that

|∇⁡(GF′∪{y1,…,yn})|≥n+1,

i.e., mG⁢(F′∪{y1,…,yn})≥n+1. In particular, mG⁢(|F′|+n)≥n+1. This contradicts Lemma 6.6 for 0<ε<1-d if n is large enough. ∎

Claim 1 implies that, for every finite F⊆S, the group GF acts transitively on T/Δ⁢(GF|T). This also implies that all Δ⁢(GF)-classes contained in T have the same size. For a finite set F⊂S, let k⁢(F) denote this size. By Corollary 6.10, we have G∈(𝒢exp+)k for some k. This implies that k⁢(F)≤k for every finite subset F of S. This also implies that there exists a finite set F so that k⁢(F) is maximal. So let us choose F⊂S so that k⁢(F) is maximal, and similarly to above, let L:=GF|T, Z:=T/Δ(L), and let π:T→Z map each element to its Δ⁢(L)-class.

Claim 2.

For any finite F′⊂S that contains F, the group GF′ acts highly transitively on Z.

Proof of Claim 2.

Let E:=GF′|T, and let K:=E/Δ(E). By the maximality of k⁢(F), it follows that Δ⁢(GF) and Δ⁢(GF′) agree on T; hence T/Δ⁢(E)=T/Δ⁢(L), and thus K≤Sym⁡(Z). As in the proof of Claim 1, it follows that K∈𝒢exp+, and Claim 1 implies that ∇⁡(K) is the universal congruence. So it is enough to show that Δ⁢(K) is the identity congruence. In order to show this, let us consider the relation on X,

R:={(x,y)∈T2∣(π(x),π(y))∈Δ(K)}∪{(x,x)∣x∈S}.

Then R is a congruence of GF′ with finite classes. Therefore, R is finer than Δ⁢(GF′). As Δ⁢(GF) and Δ⁢(GF′) agree on T, this is only possible if Δ⁢(K) is the identity congruence. Hence, the conditions of Lemma 6.13 hold for K, and thus, by Lemma 6.13, it follows that K is highly transitive. ∎

Now let us choose a prime p>k⁢(F), and let z1,…,zp∈Z. Let

F=S0⊂S1⊂⋯ and π-1⁢(z1)∪⋯∪π-1⁢(zp)=T0⊂T1⊂⋯

be sequences of finite subsets of S and T, respectively, so that ⋃Si=S and ⋃Ti=T. By Claim 2, the stabiliser GSi acts highly transitively on Z=T/Δ⁢(L). In particular, there is a permutation γi′∈G which fixes every element in Si and which acts on Ti/Δ⁢(L) as (z1⁢z2⁢…⁢zp). Now let γi:=(γi′)k⁢(F)!. Then γi|T0 is nontrivial, but γi|Ti∖T0=idTi∖T0.

Since the permutations γi have finitely many possible actions on the set T0, we can assume, by choosing a subsequence if necessary, that γi|T0 are the same for all i. Then the permutations γi converge to a permutation γ for which γ|S∪T∖T0 is trivial, but γ|T0 is not trivial. Since G is closed, it follows that γ∈G. This finishes the proof of the lemma. ∎

Corollary 6.21.

Let G∈Gexp+ be closed. Suppose that Δ⁢(G) is trivial and that G stabilises every class of ∇⁡(G) setwise. Let O1,…,Ol be the orbits of G. Suppose that each Oi is infinite and that X∖Oi is algebraically closed. Then G=Sym⁡(O1)×…×Sym⁡(Ol).

Proof.

Direct consequence of Lemma 6.20. ∎

Lemma 6.22.

Let G∈Gexp+ be closed and such that Δ⁢(G) is trivial. Suppose that G fixes every class of ∇⁡(G) setwise. Let O1,…,Om be the orbits of G, and suppose that each Oi is infinite. Then G=Sym⁡(O1)×…×Sym⁡(Om).

Proof.

Let I⊆{1,…,m} be minimal so that aclG⁡(S)=X for S:=⋃i∈IOi. Without loss of generality, we can assume that I={1,…,l} for some l≤m. By Lemma 6.16, the group G|S∈𝒢exp+ is closed. By the minimality of I, it follows that the sets S∖Oi, for i∈{1,…,l}, are algebraically closed with respect to G. By Corollary 6.21, it follows that G|S=Sym⁡(O1)×…×Sym⁡(Ol). By Lemma 6.13, it follows that G acts highly transitively on each orbit Oi. Thus, if l<m, then Lemma 6.18 implies that Δ⁢(G) is nontrivial. This means that l=m, and thus G=G|S=Sym⁡(O1)×…×Sym⁡(Om). ∎

In order to drop the condition that G fixes every ∇⁡(G)-class, we need the following observation about finite index subgroups of oligomorphic groups.

Proposition 6.23.

Let G be an oligomorphic permutation group on a countably infinite set X, and let H be a finite-index subgroup of G. Then

oni(H)≤[G:H]⋅oni(G).

In particular, H is oligomorphic.

Proof.

Choose elements γ1,…,γ[G:H]∈G such that G=⋃i=1[G:H]γi⁢H. If the tuples t1,t2,…,tl represent all injective n-orbits of G, then the tuples γi⁢tj for 1≤i≤[G:H] and 1≤j≤l represent all injective n-orbits of H. Therefore, oni(H)≤[G:H]⋅oni(G). ∎

Lemma 6.24.

Let G≤Sym⁡(X) be an oligomorphic permutation group, and let H be a finite-index subgroup of G. Then aclH⁡(x)⊆aclG⁡(x) for all x∈X.

Proof.

By Proposition 6.23, the permutation group H is oligomorphic. Let x∈X. We claim that the index k:=[Gx:Hx] is finite. Let K:=⋂g∈Gg-1Hg, that is, K is the kernel of the left action of G on the left cosets of H in G. Then K≤H, and |G/K|=[G:K] is finite. Then, by the second isomorphism theorem, we have

G/K⊇Gx⁢K/K≃Gx/(K∩Gx)=Gx/Kx.

In particular, [Gx:Kx]=|Gx/Kx| is finite. Since K≤H, we have Kx≤Hx. Therefore, [Gx:Hx] is also finite.

We have shown that k:=[Gx:Hx] is finite. Choose elements

γ1,…,γk∈Aut⁡(𝔄) such that Gx=⋃i=1kγi⁢Hx.

Let y∈aclH⁡(x). Since [H:Hx] is finite, Hx⁢(y) is finite. Therefore,

Gx⁢(y)=(⋃i=1kγi⁢Hx)⁢(y)=⋃i=1kγi⁢Hx⁢(y)

is finite, that is, y∈aclG⁡(x). This proves that aclH⁡(x)⊆aclG⁡(x). ∎

Lemma 6.25.

Let G be an oligomorphic permutation group on a countably infinite set X, and let H be a finite-index subgroup of G. Then Δ⁢(G)=Δ⁢(H).

Proof.

Clearly, Δ⁢(G) is a congruence of H with finite classes. Thus, we have Δ⁢(G)⊆Δ⁢(H). Now let (x,y)∈Δ⁢(H). Then y∈aclH⁡(x) and x∈aclH⁡(y) by Lemma 2.13. By Lemma 6.24, this implies that y∈aclG⁡(x) and x∈aclG⁡(y). Again by Lemma 2.13, we obtain (x,y)∈Δ⁢(G), showing that Δ⁢(G)=Δ⁢(H). ∎

Theorem 6.26.

Let G∈Gexp+ be closed such that Δ⁢(G) is trivial. Let O1,…,Om be the classes of ∇⁡(G). Then Sym⁡(O1)×…×Sym⁡(Om)⊆G.

Proof of Theorem 6.26.

Let K be the kernel of the action of G on {O1,…,Om}. Then [G:K] is finite, and thus, by Proposition 6.23, it follows that K∈𝒢exp+. Without loss of generality, we can assume that O1,…,Ol are the infinite orbits of K. Let Y=O1∪…∪Ol. Then X∖Y is finite, and K fixes each element in X∖Y. By Lemma 6.3, it follows that the group K|Y is in 𝒢exp+. By Lemma 6.25, it follows that Δ⁢(K) is trivial, and thus Δ⁢(K|Y) is also trivial. Moreover, K|Y fixes every class of ∇⁡(K|Y) setwise, and all orbits of K|Y are infinite. Hence, we can apply Lemma 6.22 and obtain that K|Y=Sym⁡(O1)×…×Sym⁡(Ol). Therefore, Sym⁡(O1)×…×Sym⁡(Om)=K⊆G. ∎

Corollary 6.27.

Let A∈Kexp+ be such that Δ⁢(A) is trivial. Then A∈R⁢(U).

Proof.

Apply Theorem 6.26 to Aut⁡(𝔄), and combine with Corollary 3.10. ∎

6.5 The general case

Lemma 6.28.

𝒦exp+⊆F⁢(R⁢(𝒰)).

Proof.

Let 𝔄∈𝒦exp+, and let us consider the factor mapping π:𝔄→𝔅, where 𝔅:=𝔄/Δ(𝔄). Recall that, by 𝔄/Δ⁢(𝔄), we denote any structure with

Aut⁡(𝔄/Δ⁢(𝔄))=Aut⁡(𝔄)/Δ⁢(𝔄).

If u,v∈B are in different orbits of Aut⁡(𝔅), then the tuples in π-1⁢(u) lie in different orbits of Aut⁡(𝔄) than the tuples in π-1⁢(v). Moreover, if u∈Bn is injective, then so are the tuples in π-1⁢(u). This means that the number of injective n-orbits of 𝔅 is at most oni⁢(𝔄), and thus 𝔅∈𝒦exp+. Then Δ⁢(𝔅) must be trivial; otherwise, Aut⁡(𝔅) has a nontrivial congruence all of whose classes are finite, contradicting the definition of Δ⁢(𝔄). By Corollary 6.27, it then follows that 𝔅∈R⁢(𝒰), and thus 𝔄∈F⁢(R⁢(𝒰)). ∎

The reverse containment holds as well.

Theorem 6.29.

𝒦exp+=F⁢(R⁢(𝒰))=R<∞⁢(F⁢(𝒰*)).

Proof.

We already know that

𝒦exp+⊆F⁢(R⁢(𝒰))(Lemma 6.28),

F⁢(R⁢(𝒰))⊆R<∞⁢(F⁢(𝒰*))(see Remark 5.8).

So we have to show that R<∞⁢(F⁢(𝒰*))⊆𝒦exp+. Proposition 2.33 implies that R<∞⁢(𝒦exp+)=𝒦exp+. Therefore, it is enough to show that F⁢(𝒰*)⊆𝒦exp+. So let 𝔅∈𝒰*, and let π:𝔄→𝔅 be a finite covering. Lemma 4.9 shows that π is strongly split. Therefore, we can assume that π is a strongly trivial covering map.

It follows from the description of trivial coverings given in Remark 4.11 that the orbit of an injective n-tuple t=(t1,…,tn) of a trivial covering of a unary structure is uniquely determined by the orbits of t1,…,tn and by the partition of the set {t1,…,tn} defined by the congruence ∼π. This means that the number of injective orbits of 𝔄 it at most mn⋅pk⁢(n), where

m is the number of orbits of Aut⁡(𝔄),
k is the maximal size of the classes of ∼π,
pk⁢(n) is the number of partitions of {1,…,n} with parts of size at most k (see Section 6.1).

Choose d>k-1k. Then, by Lemma 6.2, we have pk⁢(n)<c1⁢nd⁢n for some c1. Thus, oni⁢(𝔄)≤mn⁢c1⁢nd⁢n≤c2⁢nd⁢n for some c2. Therefore, 𝔄∈𝒦exp+. ∎

Remark 6.30.

Recall from Proposition 4.9 that every finite cover π:𝔄→𝔅 for 𝔅∈𝒰* is strongly split, and hence all structures in 𝒦exp+=R⁢(F⁢(𝒰*)) have a first-order interpretation in (ℕ;=) (Remark 4.12). Since (ℕ;=) is ω-stable and first-order interpretations preserve ω-stability, it follows that all structures in 𝒦exp+ are ω-stable.

6.6 Thomas’ conjecture for the class 𝒦exp+

Let k,m∈ℕ. Let 𝒢⁢(k,m) be the class of all oligomorphic permutation groups G for which the classes of Δ⁢(G) have size at most k and where ∇⁡(G/Δ⁢(G)) has at most m classes. Let 𝒮⁢(k,m) be the class of all structures whose automorphism group is in 𝒢⁢(k,m).

Lemma 6.31.

Let k,m∈N. Let B∈U*, let π:A→B be a finite covering, and let C be a quasi-covering reduct of B. Then C∈S⁢(k,m) if and only if A∈S⁢(k,m).

Proof.

By definition, Δ⁢(𝔄)=Δ⁢(ℭ)=∼π, and

∇⁡(Aut⁡(ℭ/Δ⁢(ℭ)))=∇⁡(Aut⁡(ℭ/Δ⁢(𝔄)))=∇⁡(Aut⁡(𝔄/Δ⁢(𝔄)))=∇⁡(𝔅).∎

Lemma 6.32.

Let k,m∈N. There are finitely many structures in Kexp+∩S⁢(k,m) up to bi-definability.

Proof.

By Theorem 6.29, we know that 𝒦exp+=(F∘R)⁢(𝒰). Proposition 5.11 implies that every structure ℭ in 𝒦exp+ is a quasi-covering reduct of a finite covering structure 𝔄 of some structure in 𝒰*. By Lemma 6.31, we know that ℭ∈𝒮⁢(k,m) if and only if 𝔄∈𝒮⁢(k,m). If 𝔄 is a trivial covering of some structure in 𝒰*, then by Theorem 5.10, it has finitely many quasi-covering reducts. Therefore, it is enough to show that there are finitely many structures in 𝒮⁢(k,m) up to bi-definability which are strongly trivial covering structures of some structure in 𝒰*.

Let 𝔅∈𝒰*, and let π:𝔄→𝔅 be a strongly trivial finite covering map. Let O1,…,Ol be the orbits of 𝔅. Then l≤m. Following Remark 4.11, we can assume without loss of generality that A=⊔i=1lFi×Oi for some finite sets Fi, and

Aut⁡(𝔄)=⊔i=1lidFi≀Sym⁡(Oi).

Since ∼π is a congruence with finite classes, it follows that |Fi|≤k. Then there are finitely many options for l, the sizes of the orbits Oi (they are all either one or infinite), and the sizes of the sets Fi, and if we fix these parameters, then the group Aut⁡(𝔄) is uniquely determined up to isomorphism. This implies that there are finitely many structures in 𝒮⁢(k,m) up to bi-definability which are a trivial covering structure of a structure in 𝒰*. ∎

Lemma 6.33.

Let A∈Kexp+, and let B∈R⁢(A). Let k be the size of the largest Δ⁢(A)-class, and let m be the number of ∇⁡(A)-classes. Then B∈S⁢(k,m).

Proof.

If R is a congruence of Aut⁡(𝔅), then it is also a congruence of Aut⁡(𝔄). Therefore, the size of every class of Δ⁢(𝔅) is at most k. Similarly, the number of ∇⁡(𝔅)-classes is at most the number of ∇⁡(𝔄)-classes. The number of ∇⁡(𝔅)-classes is an upper bound for the number of ∇⁡(𝔅/Δ⁢(𝔅))-classes. This proves the lemma. ∎

Lemma 6.32 and Lemma 6.33 immediately imply the following weak version of Thomas’ conjecture for the class 𝒦exp+.

Theorem 6.34.

Let A∈Kexp+. Then A has finitely many first-order reducts up to bi-definability.

Theorem 6.34 implies the (standard version of) Thomas’ conjecture as follows. First we state an important well-known link between infinite descending chains of first-order reducts and infinite signatures. We say that a structure 𝔅 has essentially infinite signature if there does not exist a structure 𝔅′ with finite signature such that Aut⁡(𝔅)=Aut⁡(𝔅′).

Lemma 6.35.

Let A be an ω-categorical structure. Then there exists an infinite sequence B1,B2,… of first-order reducts of A such that Aut⁡(B1)⊋Aut⁡(B2)⊋⋯ if and only if A has a reduct with essentially infinite signature.

Proof.

Assume that the reduct 𝔅=(B;R1,R2,…) of 𝔄 has essentially infinite signature. By assumption, 𝔅 and 𝔅n:=(B;R1,…,Rn) are not first-order interdefinable. Moreover, for every n∈ℕ, there exists an f⁢(n)∈ℕ such that 𝔅n and 𝔅f⁢(n) are not first-order interdefinable (otherwise, every relation in 𝔅 would be first-order definable in 𝔅n, contradicting our assumptions). So

𝔅1,𝔅f⁢(1),𝔅f⁢(f⁢(1)),…

provides an infinite strictly descending chain of first-order reducts of 𝔄.

Suppose conversely that 𝔅1,𝔅2,… is an infinite strictly descending chain of first-order reducts of 𝔄. Define ℭ as the first-order reduct of 𝔄 whose relations are precisely the relations of all the 𝔅i. Assume for contradiction that there exists a finite-signature structure ℭ′ with Aut⁡(ℭ′)=Aut⁡(ℭ). Let i∈ℕ be such that all relations used in the definitions of the relations of ℭ′ in ℭ already appear in the signature of 𝔅i. Then Aut⁡(𝔅i)=Aut⁡(ℭ′)=Aut⁡(ℭ)=Aut⁡(𝔅j) for all j≥i, contradicting the assumption that (𝔅i)i∈ℕ is strictly decreasing. ∎

Proposition 6.36.

Let A be an ω-categorical structure. Then A has finitely many first-order reducts up to interdefinability if and only if A has finitely many first-order reducts up to bidefinability.

Proof.

If 𝔅 is a first-order reduct of 𝔄 with essentially infinite signature, then 𝔅 has an infinite strictly descending chain of first-order reducts. Note that if Aut⁡(𝔅1)⊊Aut⁡(𝔅2), then for some n, there are strictly more orbits of n-tuples in Aut⁡(𝔅1) than in Aut⁡(𝔅2), so 𝔅1 and 𝔅2 are not bidefinable (if two reducts are bidefinable, then they have the same number of orbits of n-tuples for all n). So 𝔅 and 𝔄 have infinitely many first-order reducts up to bi-definability, so the statement is trivially true in this case.

Therefore, it suffices to show that every first-order reduct 𝔅 of 𝔄 with finite signature is bidefinable to at most finitely many reducts of 𝔅 up to interdefinability. The equivalence class of 𝔅 with respect to interdefinability is given by its orbits of n-tuples, for some finite n (since 𝔅 has finite signature), and thus the same holds for any structure which is bidefinable 𝔅. Since 𝔄 is ω-categorical, there are finitely many orbits of n-tuples in 𝔄, which implies that there are finitely many first-order reducts of 𝔄 up to interdefinability that are bidefinable with 𝔅. ∎

Theorem 6.37.

Let A∈Kexp+. Then A has finitely many first-order reducts.

Proof.

Follows from Theorem 6.34 and Proposition 6.36.∎

Corollary 6.38.

𝒦exp+ contains countably many structures up to interdefinability. It contains no structure with essentially infinite signature.

Proof.

The first statement is implied by Lemma 6.32 in combination with Proposition 6.36. The second statement follows from Theorem 6.37 and Lemma 6.35. ∎

7 Exponential orbit growth and reducts of unary structures

In this section, we show that 𝒦exp=R⁢(𝒰).

Lemma 7.1.

R⁢(𝒰)⊆𝒦exp.

Proof.

Let 𝔄∈𝒰, and let O1,…,Ok be the orbits of Aut⁡(𝔄). Then we have Aut⁡(𝔄)=∏i=1kSym⁡(Oi) by Lemma 3.1. Hence, the number of injective n-orbits of Aut⁡(𝔄) is at most kn. This implies that 𝔄∈𝒦exp, and hence 𝒰⊆𝒦exp. The statement follows from the fact that the class 𝒦exp is closed under taking first-order reducts. ∎

Lemma 7.2.

Let A∈Kexp. Then Δ⁢(A) is trivial on each infinite orbit of Aut⁡(A).

Proof.

Let us apply Lemma 6.7 for G=Aut⁡(𝔄), R=Δ⁢(𝔄), some c such that 0<c<12, and k=2. We obtain that Δ⁢(𝔄) has at most finitely many classes of size at least 2. If O is an infinite orbit of Aut⁡(𝔄), then every class of Δ⁢(𝔄) contained in O has the same size. Therefore, Δ⁢(𝔄) must be trivial on each infinite orbit O. ∎

Lemma 7.3.

Let A∈Kexp. Then we have that A is a first-order reduct of some structure B∈Kexp for which Δ⁢(B) is trivial.

Proof.

Let F be the union of finite orbits of 𝔄. Then we have that F is finite. Let 𝔅 be a structure obtained from 𝔄 by adding a constant for each element of F. Then Aut⁡(𝔅)=Aut⁡(𝔄)|F, and it is easy to see that Aut⁡(𝔄)|F∈𝒢exp. Therefore, 𝔅∈𝒦exp. Now let C be a class of Δ⁢(𝔅). If C is contained in a finite orbit of 𝔄, then by definition, |C|=1, and if C is contained in an infinite orbit of 𝔄, then |C|=1 by Lemma 7.2. Therefore, Δ⁢(𝔅) is trivial. ∎

Theorem 7.4.

𝒦exp=R⁢(𝒰).

Proof.

The containment “⊇” is Lemma 7.1. Now assume that 𝔄∈𝒦exp. By Lemma 7.3, 𝔄 is a first-order reduct of some 𝔅∈𝒦exp⊆𝒦exp+ such that Δ⁢(𝔅) is trivial. By Corollary 6.27, we obtain that 𝔅∈R⁢(𝒰). Hence,

𝔄∈(R∘R)⁢(𝒰)=R⁢(𝒰).∎

8 Additional descriptions of the classes 𝒦exp and 𝒦exp+

In this section, we present additional descriptions of the class 𝒦exp+ that follow from our main results.

8.1 Generating 𝒦exp+ from 𝒮

In this subsection, we show that 𝒦exp+ is the smallest class that contains 𝒮 and is closed under taking first-order reducts, finite covering structures, and adding constants.

Lemma 8.1.

The following inclusions hold.

𝒰*⊆(R∘F∘C)⁢(𝒮).
𝒰nf⊆(R∘F)⁢(ℕ).

Proof.

Let 𝔄∈𝒰*, and let O1,…,Om be the orbits of Aut⁡(𝔄) so that

O1={y1},…,Ol={yl}

are the finite orbits. Let F={y1,…,yl}. Pick a bijection bi between Oi and ℕ for each i∈{l+1,…,m}. Let b=⋃i=l+1mbi, and let

E:={(x,y)|x,y∈⋃i=l+1mOi,b(x)=b(y)}.

Let ℭ be the structure 𝔄 expanded by the relation E. Then

Δ(ℭ)=E∪{(x,x)∣x∈F} and Aut(ℭ/Δ(ℭ))=Sym(C/Δ(ℭ))F.

Therefore, ℭ/Δ⁢(ℭ)∈C⁢(𝒮), which shows (1). If 𝔄 has no finite orbits, then l=0 and ℭ/Δ⁢(ℭ) is bi-definable with ℕ, which shows (2). ∎

Lemma 8.2.

The classes Kexp and Kexp+ are closed under C.

Proof.

We need to show that if a permutation group G on X is in 𝒢exp or in 𝒢exp+, then so is GF for any finite F⊂X. In the case of 𝒢exp, this is clear. For the class 𝒢exp+, this is stated in Lemma 6.3. ∎

Lemma 8.3.

For any class C of structures,

(8.1)(F∘R)⁢(𝒞)⊆(R∘F)⁢(𝒞),

(F∘R<∞)⁢(𝒞)⊆(R<∞∘F)⁢(𝒞).

Proof.

Let ℭ be a structure, let 𝔅 be a first-order reduct of ℭ, and let π:𝔄→𝔅 be a finite cover. Let G:=Aut(𝔄)∩μπ-1(Aut(ℭ)). Then G is closed. So G is the automorphism group of some first-order expansion 𝔇 of 𝔄, and π:𝔇→ℭ is a finite cover. Hence, 𝔄∈(R∘F)⁢(ℭ). Moreover, if ℭ is ω-categorical and [Aut⁡(𝔅):Aut⁡(ℭ)] is finite, that is, Aut⁡(𝔅)=g1⁢Aut⁡(ℭ)∪…∪gn⁢Aut⁡(ℭ) for some g1,…,gn∈Aut⁡(𝔅), then Aut⁡(𝔄)=h1⁢Aut⁡(ℭ)∪…∪hn⁢Aut⁡(ℭ) for some hi so that μπ⁢(hi)=gi. In particular, [Aut⁡(𝔄):Aut⁡(𝔇)] is finite. ∎

If 𝔄 has no finite orbits, then every first-order reduct of 𝔄 does not have finite orbits, too, so R⁢(𝒞nf)⊆(R⁢(𝒞))nf for any class 𝒞. For 𝒞=𝒰, we even get that

(8.2)R⁢(𝒰nf)=R⁢(𝒰)nf

(see Corollary 3.7). Since a finite covering structure 𝔄 of 𝔅 has finite orbits if and only if 𝔅 has finite orbits, we have for any class 𝒞 of structures that

(8.3)F⁢(𝒞nf)=(F⁢(𝒞))nf.

Theorem 8.4.

The following equalities hold.

𝒦exp+=(R∘F∘C)⁢(𝒮).
(𝒦exp+)nf=(R∘F)⁢(ℕ).

Proof.

Clearly, ℕ∈𝒮⊂𝒦exp⊂𝒦exp+, and 𝒦exp and 𝒦exp+ are closed under R. The closure of 𝒦exp+ under F follows from 𝒦exp+=(F∘R)⁢(𝒰*) (see Theorem 6.29), and the closure under C is stated in Lemma 8.2. Finally, (𝒦exp+)nf is closed under R since R⁢((𝒦exp+)nf)⊆(R⁢(𝒦exp+))nf=(𝒦exp+)nf. This shows the inclusions ⊇ in (1) and in (2). For the converse containments, observe that

𝒦exp+=(F∘R)⁢(𝒰*)(by Theorem 6.29)

⊆(F∘R∘R∘F∘C)⁢(𝒮)(by Lemma 8.1 (1))

=(F∘R∘F∘C)⁢(𝒮)

⊆(R∘F∘F∘C)⁢(𝒮)(by (8.1) in Lemma 8.3)

=(R∘F∘C)⁢(𝒮),

which shows (1). Moreover, to show (2),

(𝒦exp+)nf=((F∘R)⁢(𝒰*))nf(by Theorem 6.29)=F⁢((R⁢(𝒰))nf)(by (8.3))=(F∘R)⁢(𝒰nf)(by (8.2))⊆(F∘R∘R∘F)⁢(ℕ)(by Lemma 8.1 (2))⊆(R∘F)⁢(ℕ)(as above).∎

8.2 Model-complete cores

The model-complete core of an ω-categorical structure has already been defined in the introduction. In this section, we show that 𝒦exp+ is the smallest class of structures that contains ℕ and is closed under taking first-order reducts, finite covers, and model-complete cores.

Lemma 8.5.

Let A be an ω-categorical structure and B its model-complete core. Then on⁢(B)≤on⁢(A) and oni⁢(B)≤oni⁢(A) for all n∈N.

Proof.

For on, this is [9, Proposition 3.6.24]. The statement for oni can be shown analogously. ∎

Corollary 8.6.

The classes Kexp and Kexp+ are closed under M.

Remark 8.7.

Analogous statements hold for the model companion instead of the model-complete core.

Definition 8.8.

Let 𝔄 be a structure with signature τ, and let F⊆A. Then let 𝔄⁢(F) denote the following τ-structure.

The domain of 𝔄⁢(F) is A(F):=(F×ℕ)⊔((A∖F)×{0}).
For each R∈τ of arity k, the relation R𝔄⁢(F) is defined as
{((x1,n1),…,(xk,nk))∣(x1,…,xn)∈R}.

Remark 8.9.

The map f:A⁢(F)→A defined by (x,n)↦x is a homomorphism from 𝔄⁢(F) to 𝔄. Conversely, the mapping g:A→A⁢(F) defined by x↦(x,0) is a homomorphism from 𝔄 to 𝔄⁢(F) (in fact, it is an embedding). Therefore, 𝔄 and 𝔄⁢(F) are homomorphically equivalent.

Remark 8.10.

It follows directly from the definition that if 𝔄 is a first-order reduct of 𝔅, and F⊆A, then 𝔄⁢(F) is a first-order reduct of 𝔅⁢(F) (since we can use the same definitions).

Lemma 8.11.

Let A∈U*, and let F be the union of the finite orbits of A. Then A⁢(F)∈Unf.

Proof.

Let O1,…,Ok be the orbits of Aut⁡(𝔄). Then Aut⁡(𝔄)=∏i=1kSym⁡(Oi) by Lemma 3.1. Let O1={y1},…,Ol={yl} be the finite orbits of Aut⁡(𝔄). Then Aut⁡(𝔄⁢(F))=∏i=1lSym⁡({yi}×ℕ)×∏i=l+1kSym⁡(Oi×{0}) has no finite orbits, and therefore 𝔄⁢(F)∈𝒰nf. ∎

Lemma 8.12.

Let A∈R⁢(U), and let F be the union of the finite orbits of A. Then A⁢(F)∈R⁢(U)nf.

Proof.

Let C1,…,Cn be the classes of ∇⁡(𝔄). Then ∏i=1nSym⁡(Ci)⊆Aut⁡(𝔄) by Lemma 3.3, and hence 𝔄 is a first-order reduct of 𝔅∈𝒰*. Lemma 8.11 implies that 𝔅⁢(F)∈𝒰nf. Remark 8.10 implies that 𝔄⁢(F) is a first-order reduct of 𝔅⁢(F). Hence, 𝔄⁢(F)∈R⁢(𝒰nf)=R⁢(𝒰)nf. ∎

Corollary 8.13.

Every structure A∈R⁢(U) is interdefinable with a model-complete core of a structure in R⁢(U)nf, i.e., R⁢(U)⊆M⁢(R⁢(U)nf).

Proof.

Let 𝔄* be the expansion of 𝔄 by all first-order definable relations. Then 𝔄* is a model-complete core and interdefinable with 𝔄. Let F be the union of the finite orbits of Aut⁡(𝔄*)=Aut⁡(𝔄). By Lemma 8.12, we know that 𝔄*⁢(F)∈R⁢(𝒰)nf. Since 𝔄* and 𝔄*⁢(F) are homomorphically equivalent, it follows that 𝔄* is the model-complete core of 𝔄*⁢(F). Hence, 𝔄∈M⁢(R⁢(𝒰)nf). ∎

Corollary 8.14.

Let B∈R⁢(U*), and let π:A→B be a finite cover. Then A is interdefinable with a model-complete core of a structure in F⁢(R⁢(U*))nf, i.e.,

F⁢(R⁢(𝒰*))⊆M⁢(F⁢(R⁢(𝒰*))nf).

Proof.

As in the previous proof, let 𝔄* be the expansion of 𝔄 by all relations that are first-order definable in 𝔄, and let F be the union of the finite orbits of Aut⁡(𝔄)=Aut⁡(𝔄*). Then π⁢(F) is the union of finite orbits of Aut⁡(𝔅). By Corollary 8.12, we know that 𝔅⁢(π⁢(F))∈R⁢(𝒰). Let π′:A⁢(F)→B⁢(π⁢(F)) be defined as

π′(x,n):={(π⁢(x),n)if⁢x∈F,(π⁢(x),0)otherwise.

Then it is easy to see that π′:𝔄⁢(F)→𝔅⁢(π⁢(F)) is a finite covering map. Hence, 𝔄⁢(F)∈F⁢(R⁢(𝒰*)). By Lemma 8.11, the structure 𝔄⁢(F) has no finite orbits, and as before, we can conclude that 𝔄 is the model-complete core of 𝔄⁢(F). ∎

Lemma 8.15.

The following identities hold.

𝒦exp=M⁢((𝒦exp)nf).
𝒦exp+=M⁢((𝒦exp+)nf).
𝒦exp+=(M∘R∘F)⁢(ℕ).

Proof.

The containments “⊇” in item (1) and item (2) follow from Corollary 8.6. By Theorems 7.4 and 6.29, we know that 𝒦exp=R⁢(𝒰) and 𝒦exp+=F⁢(R⁢(𝒰)). Then the containments “⊆” in item (1) and item (2) follow from Corollaries 8.13 and 8.14. To show item (3), observe that

𝒦exp+=M⁢((𝒦exp+)nf)(by item (2) of the lemma)=M⁢(R⁢(F⁢(ℕ)))(by item (2) of Theorem 8.4)∎

8.3 Summary

The following theorem summarises some of the equivalent characterisations of the classes 𝒦exp,𝒦exp+,(𝒦exp)nf,(𝒦exp+)nf.

Theorem 8.16.

(8.4)𝒦exp=R⁢(𝒰)=R<∞⁢(𝒰*),

(8.5)(𝒦exp)nf=R⁢(𝒰nf)=R<∞⁢(𝒰nf),

(8.6)𝒦exp+=(F∘R)⁢(𝒰)=(R∘F)⁢(𝒰),=(F∘R<∞)⁢(𝒰*)=(R<∞∘F)⁢(𝒰*)=(R∘F∘C)⁢(𝒮)=(M∘R∘F)⁢(ℕ),

(8.7)(𝒦exp+)nf=(R∘F)⁢(ℕ)=(F∘R)⁢(𝒰nf)=(F∘R<∞)⁢(𝒰nf)=(R<∞∘F)⁢(𝒰nf).

Proof.

(8.4): Corollary 3.4 states that R⁢(𝒰)=R<∞⁢(𝒰*) and Theorem 7.4 that 𝒦exp=R⁢(𝒰).

(8.5): We have R<∞⁢(𝒰nf)⊆R⁢(𝒰nf)=R⁢(𝒰)nf=(𝒦exp)nf by (8.2) and (8.4), and R⁢(𝒰)nf⊆R<∞⁢(𝒰nf) can be shown as in the proof of Corollary 3.4.

(8.6): By Theorem 6.29, we know that

𝒦exp+=(F∘R)⁢(𝒰)=(R<∞∘F)⁢(𝒰*).

This also implies that the class 𝒦exp+ is closed under F, and it is obviously closed under R, so

𝒦exp+=(R<∞∘F)⁢(𝒰*)⊆(R∘F)⁢(𝒰)⊆𝒦exp+.

The equality (F∘R)⁢(𝒰)=(F∘R<∞)⁢(𝒰*) follows from R⁢(𝒰)=R<∞⁢(𝒰*) (Corollary 3.4). The equality 𝒦exp+=(R∘F∘C)⁢(𝒮) is item (1) of Theorem 8.4, and the equality 𝒦exp+=(M∘R∘F)⁢(ℕ) is item (3) of Lemma 8.15.

(8.7): The proof of Theorem 8.4 (2) shows the following equalities:

(𝒦exp+)nf=(F∘R)⁢(𝒰nf)=(R∘F)⁢(ℕ).

Finally,

(𝒦exp+)nf=(F∘R)⁢(𝒰nf)=(F∘R<∞)⁢(𝒰nf)(as in Corollary 3.4)

⊆(R<∞∘F)⁢(𝒰nf)(by (8.1))

⊆(𝒦exp+)nf,

and thus (𝒦exp+)nf=(F∘R<∞)⁢(𝒰nf)=(R<∞∘F)⁢(𝒰nf). ∎

9 Consequences for constraint satisfaction

In the introduction, we have already mentioned that, for finite structures 𝔄, there is a complexity dichotomy for CSP⁡(𝔄): these problems are in P or NP-complete. Such a complexity dichotomy has also been conjectured for the much larger class of first-order reducts of finitely bounded homogeneous structures. A structure 𝔅 with finite relational signature τ is called finitely bounded if there exists a finite set of finite τ-structures ℱ such that a finite τ-structure 𝔄 embeds into 𝔅 if and only if no structure from ℱ embeds into 𝔄. For first-order reducts of finitely bounded homogeneous structures, there is also a more specific infinite-domain tractability conjecture [21]: assuming that 𝔄 is a model-complete core, the conjecture says that CSP⁡(𝔄) is in P if and only if 𝔄 has a pseudo-Siggers polymorphism (for a definition of pseudo-Siggers polymorphisms and a proof that the conjecture can be phrased like this, see [6]).

Let 𝔄 be a structure from 𝒦exp+ with finite relational signature. The next lemma shows that the question whether CSP⁡(𝔄) is in P or NP-complete falls into the scope of this conjecture.

Lemma 9.1.

Every structure in Kexp+ is a first-order reduct of a finitely bounded homogeneous structure.

Proof.

Let 𝔄∈𝒦exp+. By Theorem 6.29, we have 𝒦exp+=R⁢(F⁢(𝒰*)), so 𝔄 is a first-order reduct of a structure 𝔄′∈F⁢(𝒰*). By Proposition 4.9, every finite cover of a structure in 𝒰* is strongly split, so we can assume that 𝔄′ is a strongly trivial covering structure of a structure 𝔅∈𝒰*. Let ℭ be the structure from the proof of Lemma 5.4, and let τ:=({Ui,s∣i≤k,s∈Fi}∪{∼π}) be the signature of ℭ. Then it is easy to specify a finite set of forbidden finite τ-structures such that, in any finite τ-structure that avoids these structures,

the relation ∼π is an equivalence relation,
the sets denoted by the unary relations Ui,s are pairwise disjoint and cover all of C,
for all i,s, if x∼πy and x,y∈Ui,s, then x=y,
for all i,s, the cardinality of Ui,j is at most the cardinality of Ui,j in ℭ.

These are precisely the finite structures that embed into ℭ. ∎

Let 𝔄 be a structure from R⁢(F⁢(𝒰)). In this section, we discuss the consequences of our results for classifying the computational complexity of CSP⁡(𝔄). First, since R⁢(F⁢(𝒰))=𝒦exp+ is closed under M as discussed above, we can assume that 𝔄 is a model-complete core. The following lemma shows that we can even assume that 𝔄∈F⁢(𝒰).

Lemma 9.2.

Let A∈R⁢(F⁢(U)). Then there exists a model-complete core C in F⁢(U*) such that

CSP⁡(𝔄) and CSP⁡(ℭ) are polynomial-time equivalent,
the ∇⁡(ℭ)-classes are the orbits of Aut⁡(ℭ) and they are primitively positively definable in ℭ.

Proof.

Let ℭ′ be the model-complete core of 𝔄. Then ℭ′ is in

M⁢(R⁢(F⁢(𝒰)))=𝒦exp+=F⁢(R⁢(𝒰)).

So suppose that π:ℭ′→𝔅′ is a finite covering for 𝔅′∈R⁢(𝒰). Add a constant c from each ∇⁡(ℭ′)-equivalence class to ℭ′, and let ℭ be the resulting structure. Then ℭ is still ω-categorical and a model-complete core. Moreover, ℭ and 𝔄 are polynomial-time equivalent [8].

Add a constant π⁢(c) to 𝔅′ for each of the new constants c, and let 𝔅 be the structure obtained in this way. The proof of Corollary 3.5 shows that 𝔅∈𝒰*. Then π:ℭ→𝔅 is a finite cover. Therefore, ℭ∈F⁢(𝒰*). Moreover, the ∇⁡(ℭ)-classes are the orbits of ℭ, and orbits in model-complete cores are primitive positive definable [8]. ∎

It can be shown using the universal-algebraic approach to constraint satisfaction that if ℭ∈F⁢(R⁢(𝒰)), then CSP⁡(ℭ) is either in P or NP-complete. This lies beyond the scope of this article, but will appear elsewhere.

10 Conclusion and open problems

Our results imply that all structures in 𝒦exp+ are ω-stable (Remark 6.30), that they are first-order reducts of finitely bounded homogeneous structures (Proposition 9.1), and that they satisfy Thomas’ conjecture (Corollary 6.37). Do ω-stable homogeneous structures with finite relational signature in general satisfy Thomas’ conjecture, i.e., do they have finitely many reducts up to interdefinability? Note that if we drop the assumption about having a relational signature, then the answer is “no” even if we insist on 𝔄 being still ω-categorical and ω-stable (this follows from the example given in [23], which is the expansion of the countably infinite dimensional vector space over the two-element field with one non-zero constant).

Answering the question of the previous paragraph might be very ambitious, so we propose to first study a more concrete and fundamental class of structures. Let 𝒦= be the class of all structures with a first-order interpretation over (ℕ;=) (which we have already discussed in Section 1.2). Do the structures in 𝒦= satisfy Thomas’ conjecture? Is the model companion of a structure in 𝒦= also in 𝒦=? We ask the same question for the model-complete cores of structures in 𝒦=.

By our results, structures from 𝒦exp+ can be represented on a computer as follows. First, every trivial covering 𝔄 of a structure 𝔅∈𝒰 is interdefinable with a homogeneous structure in a finite relational signature ℭ (Proposition 5.4) and is finitely bounded (Lemma 9.1). So we can represent ℭ up to isomorphism by specifying these bounds. Second, finite-signature first-order reducts of ℭ can be represented by listing formulas for the relations of the reduct (we can assume that these formulas are quantifier-free since ℭ is homogeneous in a finite relational signature and hence has quantifier elimination), and storing these together with the representation for ℭ. We now ask which of the following problems are algorithmically decidable.

Given two structures in (R∘F)⁢(𝒰), decide whether they are isomorphic.
Given two structures in (R∘F)⁢(𝒰), decide whether they are interdefinable.
Given two structures in (R∘F)⁢(𝒰), decide whether they are bi-interpretable.

Szymon Toruńczyk (personal communication) observed that the first of these questions (about deciding isomorphism of two given structures) is in the larger setting of reducts of finitely bounded homogeneous structures equivalent to an open problem about decidability of first-order definability from [22] (the final open problem mentioned there).

Communicated by Christopher W. Parker

Funding source: European Research Council

Award Identifier / Grant number: 681988

Funding statement: The authors have received funding from the European Research Council (Grant Agreement no. 681988, CSP-Infinity).

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Received: 2018-12-11

Revised: 2020-12-11

Published Online: 2021-01-20

Published in Print: 2021-07-01