A counterexample to reconstruction from monoids

A counterexample to the reconstruction of $ω$-categorical structures from their endomorphism monoids

Abstract.

We present an example of two countable -categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids – in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable -categorical structure in a finite relational language which can neither be reconstructed up to first-order bi-interpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.

The first author has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039). The third author has been funded through project P27600 of the Austrian Science Fund (FWF). The fourth author has been funded through projects I836-N23 and P27600 of the Austrian Science Fund (FWF)

1. Introduction and the Result

How much information about a structure is coded into its automorphisms group ? Classical model theory provides a strong form of reconstruction of from when is big in the sense that it has for all only finitely many orbits in the componentwise action on -tuples of elements of ; such permutation groups are called oligomorphic. By the theorem of Ryll-Nardzewski, the automorphism group of a countable structure is oligomorphic if and only if is -categorical, that is, all countable models of the first-order theory of are isomorphic. The classical reconstruction result for an -categorical structure states that when is equal to as a permutation group for some structure , then has a first-order definition in , and vice versa: the two structures are first-order interdefinable. The assumption that is -categorical is in some sense best possible for this type of reconstruction: it can be seen that when has a countable signature, then the above reconstruction statement holds if and only if is -categorical.

The situation is more complicated when we only know that and are isomorphic as groups. To approach this question, it is essential to first examine and as topological groups, equipped with the topology of pointwise convergence. With this topology, automorphism groups of countable structures are precisely the closed subgroups of the full symmetric group on . A result due to Coquand (see [AZ86]) says that when and are isomorphic as topological groups (that is, via an isomorphism that is also a homeomorphism), then and are first-order bi-interpretable. We do not require the notions of interpretability and bi-interpretability here, and refer to [AZ86] for details, but mention that these notions are central in model theory since most model-theoretic concepts are stable under bi-interpretability. Hence, we focus on a subproblem: is it true that when and are isomorphic as groups, then they are also isomorphic as topological groups?

Rather surprisingly, isomorphisms between automorphism groups of countable structures are typically homeomorphisms. And in fact, it is consistent with ZF + DC that all homomorphisms between closed subgroups of are continuous, and that all isomorphisms between closed subgroups of are homeomorphisms; see the end of Section 3.2 for more explanation. Using the existence of non-principal ultrafilters on , it is relatively easy to show that there are oligomorphic permutation groups with non-continuous homomorphisms to . But it was open for a while whether for countable -categorical structures and the existence of an isomorphism between and implies the existence of an isomorphism which is additionally a homeomorphism. This problem was solved by the second author and Hewitt [EH90], by giving two structures and for which the answer was negative.

Natural objects that carry more information about a structure than are its endomorphism monoid , which consists of the set of homomorphisms from to , or, even more generally, its polymorphism clone , which consists of the set of all homomorphisms from to , for all . We are going to show the following theorem related to results of Lascar [Las89]; see also the discussion in Section 4.

Theorem 1.1.

There are countable -categorical structures , such that and are isomorphic, but not topologically isomorphic.

In fact, the two endomorphism monoids of the structures and will be the closures in of the two automorphism groups which are isomorphic, but not topologically isomorphic, presented in [EH90]. Ironically, it is its non-continuity which makes the extension of the isomorphism between those groups to their closures non-trivial, giving rise to the present work.

It has been asked in [BPP] whether there are -categorical structures whose polymorphism clones are isomorphic, but not topologically. Theorem 1.1 immediately implies a positive answer to this question: any two structures whose polymorphism clones consist essentially (that is, up to adding of dummy variables) of the functions in and , respectively, are examples.

Corollary 1.2.

There are countable -categorical structures , such that and are isomorphic, but not topologically isomorphic.

The construction in [EH90] is based on a representation of profinite groups as quotients of oligomorphic groups, due to Hrushovski, and on a non-reconstruction result for profinite groups which uses the axiom of choice. The non-reconstruction lifts to the oligomorphic groups representing the profinite groups.

In the present paper we show that it lifts further to the closures of the oligomorphic groups. The method of embedding profinite groups into quotients of oligomorphic structures is quite powerful and might be useful in different contexts as well.

The structures constructed in our proof of Theorem 1.1 have an infinite relational language. We use a well-known construction due to Hrushovski to encode countable -categorical structures into structures with a finite relational language, and show that this encoding is compatible with our examples, roughly because the encoding preserves model-completeness. That way, we obtain the following main theorem of the present article.

Theorem 1.3.

There exists a countable -categorical structure in a finite relational language such that none of , , and have reconstruction (cf. [BPP]): that is, there exists a countable -categorical structure such that and , and , and and are isomorphic, but not topologically isomorphic.

2. Preliminaries

A topological monoid is a monoid together with a topology on such that the multiplication is a continuous function. A topological group is a group such that is a topological monoid and also is continuous.

Every permutation group on a set (and, likewise, every transformation monoid on ) gives rise to a topological group (a topological monoid) as follows. We equip with the discrete topology, and with the product topology. Then composition of transformations in , and composition and taking the inverse of permutations in are continuous with respect to the subspace topology inherited from . We write for the set of all permutations of the set . If is a permutation group on a set and , the (pointwise) stabilizer of in is denoted by .

A transformation monoid is closed in if and only if it is the endomorphism monoid of a relational structure. Likewise, a permutation group is closed in if and only if it is the automorphism group of a structure with domain . The topological groups that arise in this way as automorphism groups of countable structures are precisely those Polish groups that have a compatible left-invariant ultrametric [BK96].

For a subgroup of we write , and we write for the (left-) coset of in containing . We denote by the set of all cosets of in . If is a normal subgroup of then carries a natural group structure which is a topological group with respect to the quotient topology. We write if and are isomorphic as groups, and if and are topologically isomorphic, that is, there exists an isomorphism which is also a homeomorphism. When forming direct products of topological groups and , then the group is equipped with the product topology of and .

For background on profinite groups, we refer to the text book of Ribes and Zalesskii [RZ00].

Function clones are the multivariate generalisation of transformation monoids. For a fixed set , the largest function clone on is the set , and a function clone (on ) is a subset of (called the operations) that contains all the projection maps and that is closed under composition. Each set is equipped with the product topology (again, is taken to be discrete), and then carries the sum topology. With respect to this topology, composition of operations is continuous, and the clones that are closed subsets of are precisely the polymorphism clones of structures with domain .

A clone homomorphism from a function clone to a function clone is a map from the operations of to the operations of such that for all we have . A clone isomorphism is a bijective clone homomorphism. We refer the reader to [BPP] for a more thorough treatment of function clones and topological clones.

3. The Proof

3.1. Overview

The idea is to obtain the results in the following steps.

  • There exist separable profinite groups and which are abstractly but not topologically isomorphic: but .

  • There is a oligomorphic permutation group on a countable set such that for every separable profinite group there exists a closed permutation group such that . Furthermore can be characterized in the topological group structure of as the intersection of the open normal subgroups of finite index.

It would then be natural to continue by the following steps. However, we do not know whether (3) is true, so the argument will proceed in a less direct way, but still following the outline below.

  • For the separable profinite groups and from (1), the permutation groups and are isomorphic.

  • and cannot be topologically isomorphic, since by (2) any topological isomorphism would have to send onto itself, and so and would be topologically isomorphic, contradicting (1).

  • The isomorphism between the permutation groups and extends to their topological closures and in . However, the closed monoids and are not topologically isomorphic: otherwise we would obtain a topological isomorphism between and by restricting any topological isomorphism between and , contradicting (4).

  • The closed oligomorphic function clones containing precisely the essentially unary functions obtained from and are isomorphic by extending the isomorphism between and naturally. However, they are not topologically isomorphic as otherwise and would be topologically isomorphic as well by restricting any topological isomorphism between the functions clones to their unary sort.

  • can be encoded in a structure in a finite language such that the above arguments still work.

We remark that the steps (1)-(3) have already been discussed in [EH90], but we are going to recapitulate them for the convenience of the reader and to build on the construction in the further steps. The profinite group in (1) has been known for a long time [Wit54]. Its properties were used in [EH90] to construct the profinite group that is isomorphic, but not topologically isomorphic to it. The proof of step (2) is due to an idea of Cherlin and Hrushovski, and (7) to another idea of Hrushovski.

The biggest technical challenge is step (3), and similarly, step (5). It is worth noting that we do not know whether (3) and (5) are true in general; our proof depends on the particular structure of the group from (1). In fact, our proof will deviate from the above presentation in that we will not directly work with but with a factor thereof. We find it, however, useful to have the above schema in mind since it does reflect the general proof idea.

3.2. Profinite groups

In this section we are going to discuss the profinite group that will be the basis of our counterexample. We say a subgroup is a complement of a normal subgroup of iff and is the identity subgroup.

Proposition 3.1.

There exists a separable profinite group with the following properties:

  • has a non-trivial, finite central subgroup with a dense complement in ;

  • any complement of any finite central subgroup of is dense in .

The construction of this profinite group can be found in [EH90, Theorem 4.1], where it is also used to answer a question about relative categoricity. We remark that the same group had already been constructed in  [Wit54] in a different context, namely to provide an example of a compact separable group with a non-compact commutator subgroup.

Lemma 3.2.

Let and be as in Proposition 3.1. Then:

  • is a profinite group which is isomorphic, but not topologically isomorphic to ;

  • and are isomorphic as groups, but are not topologically isomorphic.

Proof.

Since is central we have that . Since moreover is the identity subgroup, every has a unique representation , where and . Hence every coset contains exactly one representative from . So the restriction of the quotient homomorphism to is bijective and thus an isomorphism. Since is closed, is a profinite group; in particular it is compact. By Proposition 3.1 is not closed in and therefore not compact. So and cannot be topologically isomorphic.

Since is central in , we have that is isomorphic to , and so is by the above. However, no isomorphism from to can be a topological one. Otherwise, the image of (viewed as a subgroup of in the natural embedding) would be central in and so the image of would have to be a proper dense subgroup of , by Proposition 3.1. Therefore it would not be closed, contradicting compactness. ∎

Notation 3.3.

From now on, we fix groups , , and as in Proposition 3.1. We moreover denote the isomorphism from onto which sends every class to the unique element in by .

We remark that the axiom of choice was used to show the existence of the pair of subgroups in in Proposition 3.1. This seems unavoidable: it is well-known that every Baire measurable homomorphism between Polish groups is continuous (see e.g. [Kec95]). Further the statement that every set is Baire measurable is consistent with ZF+DC ([She84]). Thus the existence of two separable profinite groups (respectively two closed oligomorphic groups) that are isomorphic, but not topologically isomorphic, cannot be proven in ZF+DC (see the discussion in [BP15]). The insufficiency of ZF+DC to construct a non-continuous homomorphism between Polish groups was already observed in [Las91].

3.3. Encoding profinite groups as factors of oligomorphic groups

The next step is to describe a given separable profinite group as a factor of two oligomorphic permutation groups. Our argument is a generalization of an argument of Cherlin and Hrushovski, which can be used to show that there are oligomorphic groups without the small index property [Las82]. A similar construction is also used in [BPP14] to show that there is an oligomorphic clone on a countable set with a discontinuous homomorphism onto the projection clone. The result also appears in [EH90].

Proposition 3.4.

There is a closed oligomorphic permutation group on a countable set such that for any separable profinite group there exists a closed permutation group such that and:

  • is a closed normal subgroup of ,

  • is the intersection of the open subgroups of of finite index,

  • .

Proof.

We first prove the proposition for the special case . Let be the language containing an -ary relation symbol for all integers . Then we consider the class of all finite -structures such that

  • for all : implies that the entries of are distinct;

  • for all : , …, form a partition of the -tuples with distinct entries.

It is easy to verify that this class is a Fraïssé-class. Thus there is a unique countable homogeneous structure whose age, i.e., its set of finite induced substructures up to isomorphism, is equal to this class. Since the number of relations of any fixed arity in is finite, is -categorical. We set to be the automorphism group of .

For every , let be the -ary relation on that holds if and only if and are members of the same partition class . By definition, the relation forms an equivalence relation on the -tuples with distinct entries that has the sets as equivalence classes. We set to be the automorphism group of . Clearly every is definable in , so . By verifying that has the extension property, one can easily see that it is a homogeneous structure.

Every function in induces a permutation on the set , for every . The action of on the disjoint union of gives us a homomorphism . The homogeneity of guarantees that every permutation on a finite subset of (respecting the arities ) is induced by an element of . This fact, together with a standard back-and-forth-argument, implies that we can obtain every permutation on the full union as the action of an element of . In other words, is surjective. Every stabilizer in of a finite subset of is an open subgroup, hence is continuous and open. The kernel of is , so we have .

Finally, we want to prove that is the intersection of the open subgroups of of finite index. It is clear that contains this intersection, since is the intersection of the preimages of all the stabilizers of , , under the action of . It remains to show that has no proper open subgroup of finite index.

Suppose that has a proper open subgroup of finite index. Since is open, there is a finite tuple of distinct elements in such that its stabilizer lies entirely in . We will obtain a contradiction by studying the actions of and on . Let and be the orbits of under these actions. Now can be partitioned into subsets of the form , where . This partition is clearly preserved under the action of . For all the following holds:

Thus the index coincides with the number of partition classes in . Since this index is greater than 1, there exists outside the class .

We next claim that there exists a tuple such that all elements of the tuple are distinct. Otherwise in every tuple with an equation holds; we will derive a contradiction. For all , pick such that for all the tuples and contain no common values. This is possible by the construction of . By our assumption, for every function in the coset , contains an element of the tuple . By the choice of the , it follows that for , the cosets and are disjoint. This is a contradiction to the finite index of in .

There exists such that the tuples , and lie in the same orbit with respect to the action of . This follows from the extension property of . So there are functions such that and . Since preserves our partition, implies that lies in . But because of also lies in the same class, which is a contradiction.

We have shown the proposition for . Let now be an arbitrary separable profinite group. As such, it is topologically isomorphic to a closed subgroup of , so without loss of generality let . We set to be the preimage of under . Clearly then . Again is the intersection of all the stabilizers of in for , implying that the intersection of all open subgroups of finite index in is contained in . Since has no proper open subgroup of finite index, they are equal. ∎

Notation 3.5.

From now on let be the oligomorphic permutation group defined in the proof of Proposition 3.4 and be its domain. Also let be the quotient mapping described in the proof.

3.4. Lifting the isomorphism to the encoding groups

Let be as in Proposition 3.1. The most natural next step in the proof might be to lift the non-topological isomorphism between and to an isomorphism between and . However, we do not know if this is possible. Instead, we will work with and the closure of in a discontinuous action as a permutation group.

As technical preparation for this, we will now provide a particular representation of the topological group as a permutation group (i.e., a topological isomorphism with a permutation group).

A representation of as a permutation group

As a separable profinite group, contains a countable sequence of open normal subgroups with trivial intersection. Since is compact, the factor groups are finite. Letting act on the disjoint union of the factor groups by translation, we obtain a topologically faithful action of , i.e., a representation of as a closed permutation group on the countable set . In particular, we then have a representation of the subgroup as a (non-closed) permutation group on .

Recall that is naturally isomorphic to , but not topologically isomorphic to it. In the following, we will pick the open normal subgroups mentioned above in such a way that the restriction of the action of to (where is missing) will still be faithful and hence isomorphic to ; however, it will be topologically isomorphic to , and in particular not topologically isomorphic to . Note that the topology on is obtained from the topology of by factorizing modulo , and hence is coarser than the topology on , making such an undertaking possible.

Our action of on will moreover have the property that its restriction to will be isomorphic (as an action) to an action of on the disjoint union of certain coset spaces of , rather than of . Hence, it can be defined from alone. In particular, since is dense in , the action of can be reconstructed from and a particular sequence of normal subgroups thereof. Note that not all of the will be open, since the action of on is not a closed permutation group. In fact, only will be non-open.

It is this particular representation of as a permutation group which will allow us to lift isomorphisms to the oligomorphic permutation groups encoding our profinite groups. Note that we use the particular structure of , e.g., the density of , to obtain the representation.

To obtain the desired open normal subgroups, we first pick a sequence of open normal subgroups of whose intersection is the identity. The sequence exists since is closed and so is profinite. We now set to be the preimage of under the quotient mapping, i.e., , for all . So each is an open normal subgroup of , and . To finish the construction, we pick an open normal subgroup of whose intersection with is the identity; this is possible, because by profiniteness contains a sequence of open normal subgroups with trivial intersection, and because is finite. Finally, we set , for all .

Notation 3.6.

We now fix and as above, and let be the mapping which sends an element of to the permutation acting on by translation with .

Lemma 3.7.
  1. is faithful and continuous;

  2. is the stabilizer of under the action ;

  3. the restriction of to is a permutation group that is topologically isomorphic to ;

  4. the actions of on (via ) and on (by translation) are isomorphic.

  5. the closure of in is isomorphic to .

Proof.
  1. The elements of the family are open normal subgroups of with trivial intersection. Thus is faithful and continuous.

  2. Since is the intersection of all , it is the stabilizer of .

  3. For every the quotient group is isomorphic to . Thus the action of on is isomorphic to the action of on , which is a representation of as permutation group since the intersection of the factors is trivial by choice of the .

  4. Since is dense in and all are open, every coset in contains an element of . Thus

    One can now easily verify that the actions of on and on are isomorphic.

  5. This follows from (4) as is dense in .

The lifting

We will now consider a discontinuous action of , similarly to the action of on in Lemma 3.7, which is discontinuous if considered as an action of rather than of : otherwise it would be closed as a permutation group, but its closure as a permutation group is topologically isomorphic to .

The quotient homomorphism from Proposition 3.4 gives rise to an action of on the cosets by simply considering the composition . If we restrict this action to then it is continuous, as the composition of continuous functions. But if we regard the action on , the action fails to be continuous, since the induced permutation group is topologically isomorphic to the non-closed .

Recall that was defined as a closed, oligomorphic permutation group on a countable set . Clearly, the combined action of on fails to be continuous. By we denote the embedding of into . Then, analogously to in the profinite case, is not closed in .

Notation 3.8.

Henceforth will denote the action of on , and the closure of in .

Figure 1 gives an overview to all the group actions we are considering.

Group acting on via properties image properties
(i) faithful, cont. closed
(ii) faithful, discont. non-closed
(iii) discontinuous non-closed
(iv) restr. of (iii) continuous closed
(v) faithful, discont. oligom., non-closed
(vi) ext. of (v) faithful, cont. oligom., closed
(vii) comb. of (iv), (vi) continuous closed
Figure 1. Group actions, some of their properties, and the permutation groups they induce.
Lemma 3.9.
  1. is a closed oligomorphic permutation group.

  2. is the semidirect product .

  3. is the intersection of the open subgroups of finite index in .

  4. .

  5. is central in and isomorphic to .

  6. .

Proof.
  1. is closed by definition. As is oligomorphic on and is finite, it follows that is oligomorphic.

  2. The restriction function of to is a continuous homomorphism . Let and let be a sequence of permutations in such that converges to in . Then converges in , since for all . By we denote its limit in . The functions and are identical on , thus . Moreover, and have trivial intersection. Therefore is the semidirect product of and .

  3. Note that is open and that it maps subgroups of finite index in to subgroups of finite index in by (2). Since by Proposition 3.4 the permutation group is the intersection of the open subgroups of finite index in , we have that contains the intersection of open subgroups of finite index in .

    For the other inclusion we remark that fixes . Therefore the restriction of to is continuous. If now had a proper open subgroup of finite index, then its preimage under would be open and of finite index in . Because of Proposition 3.4 it would be equal to , a contradiction.

  4. By considering we get a continuous surjective homomorphism of onto . This gives us a continuous action of on , by further composing with the mapping . By additionally letting act on by restriction of its domain we get a continuous action of on (Item (vii) in Figure 1).

    It is easily verified that is the kernel of the action of on . So is topologically isomorphic to the permutation group that induces on via this action. By the definition of the action, if we consider its restriction to , then it induces the same permutation group on as the action of on – this permutation group is, by Lemma 3.7, topologically isomorphic to . Since the action of is continuous and is the topological closure of we get that the permutation group it induces is topologically isomorphic to the closure of the action of on , which is in turn topologically isomorphic to . In conclusion we get that .

  5. In the action of on from (4), the stabilizer of consists precisely of the elements of ; this follows from (2) and the definition of the action. Since the permutation group induced by this action on coincides with the permutation group induced by the action of on this set, and since the stabilizer of in the latter action is isomorphic to , we get that , factored by the kernel , is isomorphic to . Hence, is isomorphic to . As is a central subgroup of , is central in .

  6. Since is a central normal subgroup, the semidirect product in (2) is a direct product. We conclude that, as groups:

Notation 3.10.

Let be any closed oligomorphic permutation group on a countable set which is topologically isomorphic to . The existence of follows from the fact that is itself such a group and that is finite.

Corollary 3.11.

The closed oligomorphic permutation groups and are isomorphic, but not topologically isomorphic.

Proof.

As we have seen in Lemma 3.9 (6), and are isomorphic as groups. Recall that is the intersection of the open subgroups of finite index in , by Lemma 3.9 (3). By Proposition 3.4, is the intersection of the open subgroups of finite index in , and hence also in . Thus any topological isomorphism from to sends onto , and hence induces a topological isomorphism between the quotients and , which is a contradiction to Lemma 3.2. ∎

3.5. Extending the isomorphism to the closures of the groups

Notation 3.12.

For a permutation group , we denote by the topological closure of in the space of all transformations on its domain, equipped with the topology of pointwise convergence.

Note that the elements of are precisely the elementary embeddings to itself of any structure whose automorphism group is . Our aim in this section is to show that the monoids and are isomorphic, but not topologically isomorphic. It is clear that and are not topologically isomorphic, since the subgroups of invertible elements and are not. It is harder to show that they are isomorphic, since there seems to be no obvious way to carry it over from the permutation groups, the problem being the non-continuity of the isomorphism. We therefore need to further study the topological monoids and and how they are related to the profinite group .

Lemma 3.13.

Let be any separable profinite group. The continuous homomorphism extends to a continuous monoid homomorphism .

Proof.

Recall that was obtained via the action of on , where consists of the equivalence classes of the relations . Every element of agrees on every finite set with an element of . Therefore the functions in preserve the equivalence relations and their negations for . Since every such relation has only finitely many equivalence classes, every element of induces a permutation on them. This action of on extends the action of and gives us the continuous monoid homomorphism . ∎

Recall the discontinuous action of on the cosets via the mapping (Item (iii) in Figure 1). With the help of we see that this action has a natural extension to . As before, the restriction of this action to is continuous, and the induced permutation group is isomorphic to . It is with the action on that we lose the continuity.

By composing the continuous function with the continuous action of on , we obtain a continuous action of on . By additionally letting act on by restriction, we get a continuous action of on which extends the action of thereon.

Similarly to the situation with , we can let act on , inducing an embedding of into the set of all transformations on which extends the group embedding from Lemma 3.9.

Lemma 3.14.
  • .

  • All elements of which stabilize (pointwise) are invertible. Hence, .

  • The action of on induces a permutation group that is equal to .

  • is isomorphic to the monoid direct product .

Proof.
  • was defined as the topological closure of in , so this is immediate.

  • The functions in are injective, so by finiteness of any element of which fixes all points of is bijective.

  • The action of on induces a permutation group that is topologically isomorphic to , by Lemma 3.9 (4). The action of on extends this action. Since all permutations induced by the action of have only finite orbits, and since the action of is continuous, every element of actually induces a permutation on . Every such permutation is in turn already induced by the action of , since the permutation group induced by this action is closed. Summarizing, the functions induced by the two actions coincide, and induce a permutation group which is topologically isomorphic to .

  • Let , and assume first that fixes pointwise. So fixes and is the identity on . Note that agrees with on (but may be non-identity on ).

    By (3), there is which agrees with on . By 3.9 we can write where and . As fix all of , the same is true of . So and therefore fixes all of . Thus agrees with and therefore with on . So (as agrees with on and fixes all of ).

    Now let be arbitrary. There exists such that and agree in their action on , by (3). Then by the preceding case, is contained in , and hence so is .

    Clearly is the trivial group and is a monoid isomorphism from to its image. As , we have the result.

Let be as in Notation 3.10.

Proposition 3.15.

The closed transformation monoids and are isomorphic, but not topologically isomorphic.

Proof.

The group is topologically isomorphic to , thus is topologically isomorphic to . By Lemma 3.14 is isomorphic to , so and are isomorphic. If they were topologically isomorphic, then also the groups of invertible elements, equal to and respectively, would be topologically isomorphic. But this contradicts Corollary 3.11. ∎

3.6. Extending the isomorphism to the function clones

When is any set of finitary functions on a given set, then there exists a smallest function clone containing it, the function clone generated by . In the special case where is a transformation monoid, this clone consists precisely of those functions which arise by adding dummy variables to the functions of the monoid. In this case, if is topologically closed, then so is the function clone generated by . Thus moving from a to the clone it generates is an algebraic procedure, in contrast to the moving from a closed permutation group to its topological closure as a transformation monoid, which is topological. It is therefore much more straightforward to extend non-topological isomorphisms between closed transformation monoids to the clones they generate. The following proposition is easy, its proof can be found in [BPP].

Proposition 3.16.

Let be transformation monoids, and let be a monoid isomorphism such that both and its inverse function send constant functions to constant functions. Then extends to an isomorphism between the function clones generated by and .

Corollary 3.17.

The function clones generated by the transformation monoids and are isomorphic, but not topologically isomorphic.

Proof.

By Propositions 3.15 and 3.16, the clones are isomorphic. Any topological isomorphism between them would yield a topological isomorphism between the monoids and by restriction to the unary sort, and hence contradict Proposition 3.15. ∎

3.7. Encoding into a finite relational language

We have shown that there are -categorical structures and whose endomorphism monoids are isomorphic, but not topologically isomorphic. The structure has an infinite signature, and it is easy to see from the theorem of Coquand, Ahlbrandt, and Ziegler [AZ86] that any structure whose automorphism group is topologically isomorphic to the one of must have an infinite signature. In this section we are going to show that there is an -categorical structure in a finite language such that its automorphism group, its endomorphism monoid and its polymorphism clone do not have reconstruction.

The key ingredient for the counterexamples of the previous sections was Proposition 3.4. It gave us an encoding of the profinite group as the quotient of an oligomorphic group and the intersection of its open subgroups of finite index. Our primary goal in this section is to construct an oligomorphic permutation group that also encodes in the above sense and can be written as the automorphism group of a structure with finite signature. We will obtain with the help of a theorem due to Hrushovski, which states that every -categorical structure is definable on a definable subset of an -categorical structure with finite signature. In Proposition 3.18 we present Hrushovski’s result and a proof sketch taken from [Hod97, Theorem 7.4.8] in order to refer to this construction later on.

Proposition 3.18.

Let be a countable -categorical structure. Then there is a finite language , containing a 1-ary predicate , and an -categorical -structure , such that the domain of is equal to the elements of satisfying and the definable relations of are exactly the definable relations of restricted to .

Proof.

We can assume that is relational with atomic relations where has arity . We can also assume that every definable relation in is equivalent to an atomic formula and that for all . In particular, has quantifier elimination and is homogeneous. Let be the language consisting of the relation symbols , , and (all 1-ary), (2-ary), and (4-ary), and let be the union of and the language of . Let be the theory in which says:

  • If for some , then all entries of satisfy ;

  • if and only if ;

  • if or , then