1. Introduction
A first-order structure
$\mathfrak {A}$
is called monadically stable iff every expansion of
$\mathfrak {A}$
by unary predicates is stable. We know that this property is also equivalent to a property called tree-decomposability introduced in [Reference Baldwin and Shelah3]. The equivalence follows from the results of [Reference Baldwin and Shelah3] and they are discussed in the introduction in [Reference Lachlan34]. Further equivalent conditions for monadic stability are given in [Reference Lachlan34].
In this article we are interested in those monadically stable structures which are
$\omega $
-categorical and countable. We denote the class of these structures by
$\mathcal {M}$
. Lachlan [Reference Lachlan34] gave a bottom-up description of the class
$\mathcal {M}$
in terms of automorphism groups. Namely he showed that a structure is contained in
$\mathcal {M}$
if and only if its automorphism group can be obtained from finite structures by a finite number of iterations of certain combinatorial constructions, called
$\omega $
-stretches and loose unions. Another interesting characterization of
$\mathcal {M}$
is given by Braunfeld [Reference Braunfeld22] in terms of orbit growth on subsets, a countable
$\omega $
-categorical structure is monadically stable if and only if it is stable and its orbit growth on subsets is slower than exponential (see Theorem 3.12 for the precise formulation).
In the present article we refine the description of automorphism groups of structures in
$\mathcal {M}$
given in [Reference Lachlan34] (Section 3). We define recursively a sequence of classes of permutation groups
$\mathcal {H}_n'\colon n\in \omega $
such that
$\mathcal {H}_n'$
is exactly the class of automorphism groups of those structures in
$\mathcal {M}$
whose Morley rank are at most n. We know from the results of [Reference Lachlan34] that every monadically stable structure has finite Morley rank. This implies the following.
Theorem 1.1. A countable structure
$\mathfrak {A}$
is monadically stable if and only if
$\operatorname {Aut}(\mathfrak {A})\in \mathcal {H}':=\bigcup _{n=0}^\infty \mathcal {H}_n'$
.
The main difference between our description and the one given in [Reference Lachlan34] is that in our description the groups in
$\mathcal {H}_n'$
have an explicit definition using the groups in
$\mathcal {H}_{n-1}'$
.
We also show that
$\mathcal {H}'$
is the smallest class of permutation groups which contains all finite domain permutation groups, is closed under isomorphism, and is closed under taking finite direct products, wreath products with
$\operatorname {Sym}(\omega )$
, and finite index supergroups. This result also gives rise to a rather simple description of the class
$\mathcal {M}$
without any mention of automorphism groups.
Theorem 1.2.
$\mathcal {M}$
is the smallest class of structures which contains the one-element pure set, is closed under isomorphism, and is closed under taking finite disjoint unions, infinite copies, and finite index first-order reducts.
1.1. Finite homogenizability and finite boundedness
A structure
$\mathfrak {A}$
is called homogeneous if every homomorphism between finitely generated substructures of
$\mathfrak {A}$
can be extended to an automorphism of
$\mathfrak {A}$
. A structure is called finitely homogenizable if it is first-order interdefinable with a homogeneous structure with a finite relational signature. In [Reference Lachlan34] Lachlan showed that all structures in
$\mathcal {M}$
are finitely homogenizable.
In the study of finitely homogeneous structures we are particularly interested in the ones which are finitely bounded, i.e., the ones whose finite substructures can be described by finitely many forbidden substructures. The precise definition can be found in Section 2.7. Lachlan [Reference Lachlan34] showed that every
$\omega $
-categorical monadically stable structure is finitely homogenizable. On the other hand, by the results of [Reference Knight and Lachlan32] we know that every stable homogeneous structure is finitely bounded. This implies that every structure in
$\mathcal {M}$
is first-order interdefinable with a finitely bounded homogeneous structure (Theorem 4.9). In Section 4 we give a direct and more elementary proof of this statement using our classification.
1.2. Connection to CSPs
Let
$\mathfrak {B}$
be a structure with a finite relational signature. Then the Constraint Satisfaction Problem (CSP) over
$\mathfrak {B}$
, denoted by
$\operatorname {CSP}(\mathfrak {B})$
, is the computational problem of deciding whether a given finite structure
$\mathfrak {A}$
with the same signature as
$\mathfrak {B}$
has a homomorphism to
$\mathfrak {B}$
. Using concepts and techniques from universal algebra, Bulatov and Zhuk proved that for finite structures
$\mathfrak {B}$
the computational complexity of
$\operatorname {CSP}(\mathfrak {B})$
satisfies a dichotomy: it is either in
$\mathbf {P}$
or it is
$\mathbf {NP}$
-complete [Reference Bulatov24, Reference Zhuk39]. The universal algebraic approach for CSPs can also be generalized for countable
$\omega $
-categorical structures. However as opposed to the finite domain case one cannot expect such a CSP dichotomy for
$\omega $
-categorical structures since in general we do not even know whether
$\operatorname {CSP}(\mathfrak {B})$
is in
$\mathbf {NP}$
. In fact we know that
$\operatorname {CSP}(\mathfrak {B})$
may be undecidable even for homogeneous structures with a finite relational signature (see for instance [Reference Bodirsky and Nešetřil17]). One way to guarantee that
$\operatorname {CSP}(\mathfrak {B})$
is at least in
$\mathbf {NP}$
is to assume that
$\mathfrak {B}$
is homogeneous and finitely bounded, or a reduct of such a structure. This motivates the following complexity dichotomy conjecture, originally presented in [Reference Bodirsky, Pinsker and Pongrácz20].
Conjecture 1.3 (Infinite domain CSP dichotomy conjecture)
Let
$\mathfrak {A}$
be a countable finitely bounded homogeneous structure, and let
$\mathfrak {B}$
be a first-order reduct of
$\mathfrak {A}$
with a finite relational signature. Then the
$\operatorname {CSP}(\mathfrak {B})$
is either in
$\mathbf {P}$
or it is
$\mathbf {NP}$
-complete.
The conjecture above has been solved for many classes of structures, such as the first-order reducts of:
-
• the pure set
$(\mathbb {N};=)$ [Reference Bodirsky and Kára14],
-
• the universal linear order
$(\mathbb {Q};<)$ [Reference Bodirsky and Kára15],
-
• the random graph [Reference Bodirsky and Pinsker18],
-
• the random poset [Reference Kompatscher and Van Pham33],
-
• unary
$\omega $ -categorical structures [Reference Bodirsky and Mottet16],
and all expansions of the homogeneous RCC5-structure by first-order definable relations [Reference Bodirsky and Bodor10]. (RCC5-structures arise in the formalism of spatial reasoning, introduced in [Reference Bennett5].)
Theorem 4.9 implies that all structures in
$\mathcal {M}$
fall into the scope of the conjecture above. We mention one more property of the class
$\mathcal {M}$
which makes it a good candidate for a CSP dichotomy result, namely that it is closed under taking model-complete cores. A countable
$\omega $
-categorical structure is called a model-complete core if the closure of its automorphism group is the same as its endomorphism monoid. Two structures are called homomorphically equivalent if there exist homomorphisms between them in both directions. Clearly, if two structures are homomorphically equivalent then they have the same CSP. We know that every
$\omega $
-categorical structure
$\mathfrak {B}$
is homomorphically equivalent to an (up to isomorphism unique) model-complete core
$\mathfrak {C}$
[Reference Bodirsky6]. Moreover the structure
$\mathfrak {C}$
is also
$\omega $
-categorical, and it is called the model-complete core of
$\mathfrak {B}$
. As we argued above, every countable
$\omega $
-categorical structures has the same CSP as its model-complete core. Moreover some of the universal algebraic techniques only work under the assumption that the domain structure is a model-complete core. Thus in the analysis of CSPs we usually prefer to work with the model-complete cores of structures. This reduction is a key step in the solution of all the dichotomy results mentioned above. For this reason when formulating a CSP dichotomy for a class
$\mathcal {C}$
of
$\omega $
-categorical structures it is useful to have a class
$\mathcal {C}$
which is closed under taking model-complete cores. This property is satisfied by
$\mathcal {M}$
for simple orbit growth reasons (see Section 3).
For more details on infinite domain CSPs we refer the reader to [Reference Barto and Pinsker4], [Reference Bodirsky7], or [Reference Bodirsky9].
1.3. Thomas’ conjecture
In [Reference Thomas37] Thomas made the conjecture that a countable homogeneous structure over a finite relational signature has only finitely many reducts up to first-order interdefinability. The conjecture has been verified for many well-known homogeneous structures. The list includes the rationals with the usual ordering [Reference Cameron25], the countably infinite random graph [Reference Thomas37], the homogeneous universal
$K_n$
-free graphs [Reference Thomas38], the expansion of
$({\mathbb Q};<)$
by a constant [Reference Junker and Ziegler31], the universal homogeneous partial order [Reference Pach, Pinsker, Pluhár, Pongrácz and Szabó35], the random ordered graph [Reference Bodirsky, Pinsker and Pongrácz19], and many more [Reference Agarwal1, Reference Agarwal and Kompatscher2, Reference Bodirsky, Bradley-Williams, Pinsker and Pongrácz12, Reference Bodirsky, Jonsson and Van Pham13]. Note that if we drop the assumption that the signature of the homogeneous structure
$\mathfrak {A}$
is relational, then Thomas’ conjecture is false even if we keep the assumption that
$\mathfrak {A}$
is
$\omega $
-categorical: already the countable atomless Boolean algebra has infinitely many first-order reducts [Reference Bodor, Cameron and Szabó21].
Even though Thomas’ conjecture has been verified for several individual structures, there does not seem to be much progress towards proving it in its full generality. In a recent paper [Reference Bodirsky and Bodor11] the authors showed that Thomas’ conjecture holds for all cellular structures. In this article we further generalize this result by showing that Thomas’ conjecture also holds for the entire class
$\mathcal {M}$
(Theorem 5.11). Finally, we mention a recent result by Simon [Reference Simon36] where he showed that Thomas’ conjecture holds for another robust class of structures, namely the class of
$\omega $
-categorical primitive NIP structures with thorn rank 1. This class contains for instance all the generalized random permutation i.e., the Fraïssé limit
$(M;<_1,\dots ,<_n)$
of all finite structures equipped with n linear orders.
2. Preliminaries
2.1. Permutation groups
For a group G we write
$H\leq G$
if H is a subgroup of G and
$H\triangleleft G$
if H is a normal subgroup of G. If
$H\leq G$
then we write
$|G:H|$
for the index of H in G. For a set X we write
$\operatorname {id}(X)$
for the identity map on X. Sometimes we drop the set X from the notation if it is clear from the context.
We say that a group G is a permutation group if G is a subgroup of
$\operatorname {Sym}(X)$
for some X. Note that whenever G is a permutation group then the set X witnessing this fact is unique (it is the domain of the identity element). We call this set the domain of G, and we denote it by
$\operatorname {Dom}(G)$
.
If
$G\subset \operatorname {Sym}(X)$
is a permutation group, and
$e\colon X\to Y$
is any injective function, then the map defined as
$\iota (e)\colon \gamma \mapsto e\gamma e^{-1}$
is a homomorphism from G to
$\operatorname {Sym}(e(X))$
, and is referred to as the homomorphism induced by e.
Two permutation groups G and H are called isomorphic as permutation groups if there is a bijection
$e\colon \operatorname {Dom}(G)\rightarrow \operatorname {Dom}(H)$
such that the homomorphism
$\iota (e)$
induced by e is an isomorphism between G and H. Note that this condition is strictly stronger than saying that G and H are isomorphic as groups. For instance the trivial permutation groups
$\{\operatorname {id}(X)\}$
are isomorphic for every set X as groups, but
$\{\operatorname {id}(X)\}$
and
$\{\operatorname {id}(Y)\}$
are isomorphic as permutation groups only if
$|X|=|Y|$
. All the groups in this article will be permutation groups, and when we say that two permutation groups are isomorphic we always mean that they are isomorphic as permutation groups.
Assume that G is a permutation group, and
$Y\subset \operatorname {Dom}(G)$
. Then we use the following notation.
-
•
$G_{Y}$ denotes the pointwise stabilizer of Y, that is,
$G_Y=\{g\in G\colon \forall y\in Y(g(y)=y)\}$ .
-
•
$G_{\{Y\}}$ denotes the setwise stabilizer of Y, that is,
$G_{\{Y\}}=\{g\in G\colon \forall y\in Y(g(y)\in Y)\}$ .
-
•
$G|_Y$ denotes the restriction of G to Y, that is,
$G|_Y=\{h|_Y\colon h\in G\}$ , provided that Y is invariant under G.
-
•
$G_{(Y)}:=G_{\{Y\}}|_Y=G_{\{\operatorname {Dom}(G)\setminus Y\}}|_Y$ .
-
•
$G_{((Y))}:=G_{\operatorname {Dom}(G)\setminus Y}|_Y$ .
Whenever the set Y is finite, say
$Y=\{y_1,\dots ,y_n\}$
, we simply write
$G_{y_1,\dots ,y_n}$
for
$G_Y$
.
If E is an equivalence relation defined on a set X then we denote by
$X/E$
the quotient set of X with respect to E, and for any
$x\in X$
we write
$[x]_E$
for its equivalence class with respect to E. Then by definition
$(x,y)\in E$
iff
$[x]_E=[y]_E$
. An equivalence relation E defined on
$\operatorname {Dom}(G)$
for some permutation group G is called a congruence of G if and only if E is preserved by every permutation in G. In this case there is a natural map
$G\to \operatorname {Sym}(\operatorname {Dom}(G)/E)$
that maps
$g\mapsto g/E$
, where
$g/E([x]_E)=[g(x)]_E$
. We define
$G/E$
as
$\{g/E\colon g\in G\}$
.
2.2. Topology
Every permutation group is naturally equipped with a topology, called the topology of pointwise convergence. This topology can be defined as the subspace topology of the product space
${}^XX$
where X is equipped with the discrete topology. A sequence
$(g_n)_n$
in G converges to a permutation
$g\in \operatorname {Sym}(X)$
with respect to this topology if and only if for all finite subsets F of X there exists an n such that
$g_i|_F=g|_F$
for all
$i\geq n$
.
We say that a permutation group
$G\leq \operatorname {Sym}(X)$
is closed if it is closed in the topology of pointwise convergence. Equivalently,
$G\leq \operatorname {Sym}(X)$
is closed if and only if it satisfies the following property: for all
$g\in \operatorname {Sym}(X)$
, if for every finite
$F\subset X$
there exists
$g'\in G$
such that
$g'|F=g|F$
then
$g\in G$
.
2.3. Orbit growth functions, oligomorphic groups
There are three natural sequences counting orbits attached to a permutation group introduced and discussed in general in [Reference Cameron26, Reference Cameron27].
Definition 2.1. Let
$G\leq \operatorname {Sym}(X)$
be a permutation group, and let n be a positive integer. Then:
-
•
$o_n(G)$ denotes the number of n-orbits of G, i.e., the number of orbits of the natural action
$G\curvearrowright X^n$ .
-
•
$o^s_n(G)$ denotes the number of orbits of n-subsets of G, i.e., the number of orbits of the natural action
$G\curvearrowright {X\choose n}=\{Y\subset X\colon |Y|=n \}$ .
A permutation group is called oligomorphic if and only if
$o_n(G)$
is finite for all n.
Remark 2.2. It follows easily from the definitions that for any permutation group G the following inequalities hold:
$o^s_n(G)\leq o^i_n(G)\leq o_n(G)\leq n!o^s_n(G)$
. Therefore the following are equivalent:
-
• G is oligomorphic.
-
•
$o^i_n(G)$ is finite for all n.
-
•
$o^s_n(G)$ is finite for all n.
We often use the following easy observation.
Observation 2.3. If
$G\leq H$
are permutation groups, then
$f_n(H)\leq f_n(G)$
where f is any of the operators
$o,o^i$
, or
$o^s$
. In particular every group containing an oligomorphic group is oligomorphic.
2.4. Direct products
Let
$\{X_i\colon i\in I\}$
be a set of pairwise disjoint sets, and for each
$i\in I$
let
$G_i$
be a permutation group acting on
$X_i$
. Then we define the direct product of the permutation groups
$G_i$
, denoted by
$\prod _{i\in I}G_i$
, to be the set of all permutations of
$X:=\bigcup \{X_i\colon i\in I\}$
which can be written as
$\bigcup \{f(i)\colon i\in I\}$
, for some
$f\colon I\to \prod _{i\in I}G_i$
that satisfies
$f(i)\in G_i$
for all
$i\in I$
. Then
$\prod _{i\in I}G_i$
as a group is also the usual direct product of the groups
$G_i\colon i\in I$
.
Let
$G\leq \operatorname {Sym}(X)$
be a permutation group and let I be any set. Then we define the power
$G^I$
to be
$\prod _{i\in I}G_s$
, where
$G_i:=\{(g,i)\colon g\in G\}$
and
$(g,i)(g',i)=(gg',i)$
. This makes sense since the sets
$\operatorname {Dom}(G_i)=X\times \{i\}: i\in I$
are pairwise disjoint. Then
$\operatorname {Dom}(G^I)=X\times I$
, and since all
$G_i$
are isomorphic to G it follows that
$G^I$
is isomorphic to the power
$G^{|I|}$
as a group.
It follows easily from the definition that any direct product of closed groups is again closed. This also means that any power of a closed group is closed.
It is easy to see that every n-orbit of a direct product
$G:=\prod _{i\in I}G_i$
can be written as the union of
$n_i$
-orbits of
$G_i$
for some sequence of natural numbers
$(n_i)_{i\in I}$
with
$\sum _{i\in I}n_i=n$
. This implies that if I is finite then the number of n-orbits of G is at most
${|I| \choose n}M^n$
where M is the maximum of
$o_n(G_i)$
with
$i\in I$
. In particular if
$o_n(G_i)$
is finite for all
$i\in I$
and I is finite then
$o_n(G)$
is also finite. Thus any finite direct product of oligomorphic groups is again oligomorphic.
2.5. Wreath products
Let
$G\leq \operatorname {Sym}(X)$
and
$H\leq \operatorname {Sym}(Y)$
be permutation groups. Then we define the wreath product of the groups G and H, denoted by
$G\wr H$
, as follows. The domain set of
$G\wr H$
is
$X\times Y$
and a permutation
$\sigma \in \operatorname {Sym}(X\times Y)$
is contained in
$G\wr H$
if and only if there exist
$\beta \in H$
and, for all
$y\in Y$
,
$\alpha _y\in G$
such that, for all
$x\in X$
and
$y\in Y$
we have
$\sigma (x,y)=(\alpha _y(x),\beta (y))$
. Note that
$\beta $
is unique given
$\sigma $
, and we denote it by
$\pi (\sigma )$
. Then the map
$\sigma \mapsto \pi (\sigma )$
defines a surjective homomorphism from
$G\wr H$
to H, and the kernel of this homomorphism is exactly
$G^Y$
. Moreover the map
$e\colon \beta \mapsto ((x,y)\mapsto (x,\beta (y))$
splits the homomorphism
$\pi $
, that is
$\pi \circ e$
is the identity map on H. This implies that
$G\wr H$
can be written as a semidirect product
$G^Y\rtimes H$
(or more precisely
$G^Y\rtimes e(H)$
).
In this document we are only interested in the case when the group H above is the full symmetric group.
Lemma 2.4. Let
$G\leq \operatorname {Sym}(X)$
be a permutation group, and let Y be a set. Then:
-
(1) if G is closed then so is
$G\wr \operatorname {Sym}(Y)$ , and
-
(2) if G is oligomorphic then so is
$G\wr \operatorname {Sym}(Y)$ .
Proof (1) A permutation
$\sigma $
of
$X\times Y$
is contained in
$\operatorname {Sym}(X)\wr \operatorname {Sym}(Y)$
if and only if it preserves the equivalence relation
$E:=\{((x_1,y_1),(x_2,y_2))\colon y_1=y_2\}$
. This implies immediately that the group
$\operatorname {Sym}(X)\wr \operatorname {Sym}(Y)$
is closed. Now let
$(\sigma _n)_n$
be a sequence in
$G\wr \operatorname {Sym}(Y)$
converging to a permutation
$\sigma \in \operatorname {Sym}(X\times Y)$
. Then there exist
$\beta _n\in \operatorname {Sym}(Y),\alpha _{y,n}\in G$
such that for all
$x\in X$
and
$y\in Y$
we have
$\sigma _n(x,y)=(\alpha _{y,n}(x),\beta _n(y))$
. Since
$\operatorname {Sym}(X)\wr \operatorname {Sym}(Y)$
is closed we know that
$\sigma \in \operatorname {Sym}(X)\wr \operatorname {Sym}(Y)$
, and thus there exist
$\beta \in \operatorname {Sym}(Y)$
and
$\alpha _y\in \operatorname {Sym}(X)$
such that for all
$x\in X$
and
$y\in Y$
we have
$\sigma (x,y)=(\alpha _y(x),\beta (y))$
. We claim that the sequence
$(\alpha _{n,y})_n$
converges to
$\alpha _y$
for all
$y\in Y$
. Indeed let F be an arbitrary finite subset of X. Then since
$(\sigma _n)_n\rightarrow \sigma $
it follows that we can choose an index
$n\in \omega $
such that for all
$i\geq n$
we have
$\sigma |_{F\times \{y\}}=(g_i)|_{F\times \{y\}}$
for all
$y\in Y$
, and thus
$\alpha _{i,y}|_F=\alpha _y|_F$
. We have shown that
$(\alpha _{n,y})_n$
converges to
$\alpha _y$
, and thus
$\alpha _y\in G$
since G is closed. This shows that
$\sigma \in G\wr \operatorname {Sym}(Y)$
.
(2) Let
$n\in \omega $
, and let Z be any n-element subset of Y. Note that in this case every n-orbit of
$G\wr \operatorname {Sym}(Y)$
contains a tuple in
$X\times Z$
. Indeed, if
$\vec {a}\in (X\times Y)^n$
then
$\vec {a}\in (X\times Z')^n$
for some
$Z'\subset Y$
with
$|Z'|\leq n$
. Let
$\beta \in \operatorname {Sym}(Y)$
such that
$\beta (Z')\subset Z$
. Then the map
$(x,y)\mapsto (x,\beta (y))$
, which is contained in
$\{\operatorname {id}(X)\}\wr \operatorname {Sym}(Y)$
, maps
$\vec {a}$
into
$(X\times Z)^n$
. Let us also observe that
$(G\wr \operatorname {Sym}(Y))_{(Z)}=G\wr \operatorname {Sym}(Z)$
. In particular two n-tuples from
$X\times Z$
are contained in the same orbit of
$G\wr \operatorname {Sym}(Z)$
if and only if they are contained in the same orbit of
$G\wr \operatorname {Sym}(Y)$
. From this it follows that
$o_n(G\wr \operatorname {Sym}(Y))=o_n(G\wr \operatorname {Sym}(Z))$
. We have already seen that the group
$G^{Z}$
is oligomorphic. On the other hand
$G\wr \operatorname {Sym}(Z)$
contains
$G^Z$
, and thus by Observation 2.3 it follows that
$G\wr \operatorname {Sym}(Z)$
is oligomorphic. In particular
$o_n(G\wr \operatorname {Sym}(Z))=o_n(G\wr \operatorname {Sym}(Z))$
is finite. Since this holds for all
$n\in \omega $
it follows that
$G\wr \operatorname {Sym}(Y)$
is oligomorphic.
2.6. Reducts of
$\omega $
-categorical structures
In this article every structure is assumed to be countable.
Notation 2.5. For a structure
$\mathfrak {A}$
we denote by
$\Sigma (\mathfrak {A})$
the signature of
$\mathfrak {A}$
.
We say that a structure is relational if its signature does not contain any function symbols.
Notation 2.6. Let
$\mathfrak {A}$
be a relational structure. Then for a relation R on
$\mathfrak {A}$
we write
$\operatorname {ar}(R)$
for the arity of R. We denote by
$m(\mathfrak {A})$
the maximal arity of relations of
$\mathfrak {A}$
.
$m(\mathfrak {A})$
is defined to be
$\infty $
if such a maximum does not exist.
Remark 2.7. In this article we always assume that the signature of any structure contains the symbol “
$=$
” (represented as equality). This means that the value of
$m(\mathfrak {A})$
is always at least 2.
Definition 2.8. Let
$\mathfrak {A}$
and
$\mathfrak {B}$
be structures. We say that
$\mathfrak {B}$
is a reduct of
$\mathfrak {A}$
iff
$\operatorname {Dom}(\mathfrak {A})=\operatorname {Dom}(\mathfrak {B})$
and every relation, function, and constant of
$\mathfrak {B}$
is first-order definable in
$\mathfrak {A}$
. Two structures are called interdefinable if they are reducts of one another.
The structures
$\mathfrak {A}$
and
$\mathfrak {B}$
are called bidefinable iff there exist a reduct
$\mathfrak {A}'$
of
$\mathfrak {A}$
with
$\Sigma (\mathfrak {A}')=\Sigma (\mathfrak {B})$
, a reduct
$\mathfrak {B}'$
of
$\mathfrak {B}$
with
$\Sigma (\mathfrak {B}')=\Sigma (\mathfrak {A})$
, and a bijection i from A to B such that i defines an isomorphism from
$\mathfrak {A}$
to
$\mathfrak {B}'$
and from
$\mathfrak {A}'$
to
$\mathfrak {B}$
.
We say that
$\mathfrak {B}$
is a finite-index reduct of
$\mathfrak {A}$
iff
$|\operatorname {Aut}(\mathfrak {B}):\operatorname {Aut}(\mathfrak {A})|$
is finite.
Lemma 2.9. The structures
$\mathfrak {A}$
and
$\mathfrak {B}$
are bidefinable if and only if
$\mathfrak {B}$
is interdefinable with a structure which is isomorphic to
$\mathfrak {A}$
.
Proof Let us assume that
$\mathfrak {A}$
and
$\mathfrak {B}$
are bidefinable, and let
$\mathfrak {B}'$
and
$\mathfrak {A}'$
be structures and let
$i\colon A\rightarrow B$
be a bijection witnessing this. Then by definition
$\mathfrak {A}$
and
$\mathfrak {B}'$
are isomorphic, and we claim that
$\mathfrak {B}$
and
$\mathfrak {B}'$
are interdefinable. To see this, it suffices to show that
$\mathfrak {B}$
is a reduct of
$\mathfrak {B}'$
. But this follows from the fact that
$\mathfrak {A}'$
is a reduct of
$\mathfrak {A}$
, and i defines an isomorphism from
$\mathfrak {A}'$
to
$\mathfrak {B}$
and from
$\mathfrak {A}$
to
$\mathfrak {B}'$
. This finishes the proof of the “only if” direction of the lemma.
Now let us assume that
$\mathfrak {B}$
and
$\mathfrak {B}'$
are interdefinable, and
$\mathfrak {B}'$
and
$\mathfrak {A}$
are isomorphic. Let i be an isomorphism from
$\mathfrak {A}$
to
$\mathfrak {B}'$
. Let us define the structure
$\mathfrak {A}'$
as follows. The domain set of
$\mathfrak {A}'$
is A,
$\Sigma (\mathfrak {A}')=\Sigma (\mathfrak {B})$
, and every symbol in
$\Sigma (\mathfrak {B})$
is realized in
$\mathfrak {A}'$
in such a way that i defines an isomorphism from
$\mathfrak {A}'$
to
$\mathfrak {B}$
. By definition
$\mathfrak {B}'$
is a reduct of
$\mathfrak {B}$
. It remains to show that
$\mathfrak {A}'$
is also a reduct of
$\mathfrak {A}$
. By definition
$\mathfrak {B}$
is also a reduct of
$\mathfrak {B}'$
. Since
$i^{-1}$
defines an isomorphism from
$\mathfrak {B}'$
to
$\mathfrak {A}$
and from
$\mathfrak {B}$
to
$\mathfrak {A}'$
, we obtain that
$i^{-1}(\mathfrak {B})=\mathfrak {A}'$
is a reduct of
$i^{-1}(\mathfrak {B}')=\mathfrak {A}$
. Thus
$\mathfrak {A}$
and
$\mathfrak {B}$
are bidefinable.
From now on we use the description given in Lemma 2.9 for bidefinability instead of its definition. The reason why we did not define bidefinability this way is that from this description it is not obvious that bidefinability is a symmetric relation.
Remark 2.10. Clearly interdefinability implies bidefinability, but the converse implication does not hold necessarily even if the structures are assumed to have the same domain set. For instance the structures
$(\mathbb {N};=,0)$
and
$(\mathbb {N};=,1)$
are bidefinable but not interdefinable.
When we are talking about reducts of a given structure
$\mathfrak {A}$
we usually consider two reducts the same if they are interdefinable. It turns out that under the assumption that
$\mathfrak {A}$
is
$\omega $
-categorical the reducts of
$\mathfrak {A}$
are completely described by the closed supergroups of the automorphism of
$\mathfrak {A}$
(see Theorem 2.13).
Definition 2.11. A countable structure
$\mathfrak {A}$
is called
$\omega $
-categorical if and only if it is isomorphic to every countable model of its first-order theory.
By the theorem of Engeler, Ryll-Nardzewski, and Svenonius, we know that a countably structure
$\mathfrak {A}$
is
$\omega $
-categorical if and only if
$\operatorname {Aut}(\mathfrak {A})$
is oligomorphic (see for instance [Reference Hodges29]). Given a structure
$\mathfrak {A}$
we use the notation
$f_n(\mathfrak {A})$
for
$f_n(\operatorname {Aut}(\mathfrak {A}))$
where n is a positive integer and f is any of the operators
$o,o^i$
, or
$o^s$
(see Definition 2.1). Then Observation 2.3 has the following analog for structures and reducts.
Lemma 2.12. Let
$\mathfrak {B}$
be a reduct of a structure
$\mathfrak {A}$
. Then
$f_n(\mathfrak {B})\leq f_n(\mathfrak {A})$
where f is any of the operators
$o,o^i$
, or
$o^s$
. In particular every reduct of an
$\omega $
-categorical structure is
$\omega $
-categorical.
We also know that for an
$\omega $
-categorical structure
$\mathfrak {A}$
a relation (or a function or a constant) defined on A is first-order definable if and only if it is preserved by every automorphism of
$\mathfrak {A}$
(see [Reference Hodges29]). This implies the following theorem.
Theorem 2.13. Let
$\mathfrak {A}$
be a countable
$\omega $
-categorical structure. Then every closed group
$\operatorname {Aut}(\mathfrak {A})\subset G\subset \operatorname {Sym}(A)$
is an automorphism group of a reduct of
$\mathfrak {A}$
, and two reducts of
$\mathfrak {A}$
have the same automorphism group if and only if they are interdefinable.
Theorem 2.13 tells us that given a countable
$\omega $
-categorical structure
$\mathfrak {A}$
the map
$\mathfrak {B}\mapsto \operatorname {Aut}(\mathfrak {B})$
defines a bijection between the reducts of
$\mathfrak {A}$
(up to interdefinability), and the closed supergroups of
$\operatorname {Aut}(\mathfrak {A})$
, that is the closed permutation groups containing
$\operatorname {Aut}(\mathfrak {A})$
.
An easy consequence of Theorem 2.13 is the following.
Lemma 2.14. Let
$\mathfrak {A}$
and
$\mathfrak {B}$
be countable structures, so that at least one of them is
$\omega $
-categorical. Then
$\mathfrak {A}$
and
$\mathfrak {B}$
are bidefinable if and only if
$\operatorname {Aut}(\mathfrak {A})$
and
$\operatorname {Aut}(\mathfrak {B})$
are isomorphic as permutation groups.
Proof We can assume without loss of generality that
$\mathfrak {A}$
is
$\omega $
-categorical.
Suppose first that
$\mathfrak {A}$
and
$\mathfrak {B}$
are bidefinable. Then by Lemma 2.9 we know that
$\mathfrak {B}$
is interdefinable with some structure
$\mathfrak {B}'$
which is isomorphic to
$\mathfrak {A}$
. Then
$\operatorname {Aut}(\mathfrak {B})=\operatorname {Aut}(\mathfrak {B}')$
and it is isomorphic to
$\operatorname {Aut}(\mathfrak {A})$
.
For the other direction let i be a bijection from A to B such that i induces an isomorphism from
$\operatorname {Aut}(\mathfrak {A})$
to
$\operatorname {Aut}(\mathfrak {B})$
. Let
$\mathfrak {A}':=i^{-1}(\mathfrak {B})$
. Then we have
$\operatorname {Aut}(\mathfrak {A}')=\iota (i^{-1})\operatorname {Aut}(\mathfrak {B})=\iota (i^{-1})(\iota (i)(\operatorname {Aut}(\mathfrak {A})))=\operatorname {Aut}(\mathfrak {A})$
. Theorem 2.13 then implies that
$\mathfrak {A}$
and
$\mathfrak {A}'$
are interdefinable. On the other hand it follows from the definition that
$\mathfrak {A}'$
is isomorphic to
$\mathfrak {B}$
. Therefore by Lemma 2.9 the structures
$\mathfrak {A}$
and
$\mathfrak {B}$
are bidefinable.
Lemma 2.14, combined with Theorem 2.13 implies that for a countable
$\omega $
-categorical structure the map
$\mathfrak {B}\mapsto \operatorname {Aut}(\mathfrak {B})$
defines a bijection between the reducts of
$\mathfrak {A}$
up to bidefinability, and the closed supergroups of
$\operatorname {Aut}(\mathfrak {A})$
up to isomorphism.
2.7. Homogeneous structures, Thomas’ conjecture
Definition 2.15. A structure
$\mathfrak {A}$
is called homogeneous if every isomorphism between finitely generated substructures of
$\mathfrak {A}$
can be extended to an automorphism of
$\mathfrak {A}$
.
Fact 2.16. Every homogeneous structure with a finite relational signature is
$\omega $
-categorical.
Proof Let
$\mathfrak {A}$
be a homogeneous structure with a finite relational signature
$\tau $
. We show that
$\operatorname {Aut}(\mathfrak {A})$
is oligomorphic. Since
$\tau $
is finite, and only contains relational and constant symbols, it follows that every finitely generated substructure is finite. Then using the finiteness of
$\tau $
again it follows that for all
$n\in \omega $
there exist only finitely many
$\tau $
-structures up to isomorphism. In particular
$\mathfrak {A}$
has finitely many quantifier-free types. It follows from the homogeneity of
$\mathfrak {A}$
that if two n-tuples have the same quantifier-free type then they are in the same orbit of
$\operatorname {Aut}(\mathfrak {A})$
. This implies that
$o_n(\mathfrak {A})$
is finite. Therefore
$\operatorname {Aut}(\mathfrak {A})$
is oligomorphic.
Thomas [Reference Thomas37] conjectured that every homogeneous structure
$\mathfrak {A}$
with a finite relational signature has only finitely many reducts up to interdefinability.
The fact that a given
$\omega $
-categorical structure has finitely many reducts can be formulated in many different ways, as the following proposition shows.
Proposition 2.17. Let
$\mathfrak {A}$
be a countable
$\omega $
-categorical structure. Then the following are equivalent:
-
(1)
$\mathfrak {A}$ has finitely many reducts up to interdefinability.
-
(2)
$\mathfrak {A}$ has finitely many reducts up to bidefinability.
-
(3) There exists some
$N\in \omega $ such that every reduct of
$\mathfrak {A}$ is interdefinable with a relational structure
$\mathfrak {B}$ with
$m(\mathfrak {B})\leq N$ .
-
(4)
$\operatorname {Aut}(\mathfrak {A})$ has finitely many closed supergroups in
$\operatorname {Sym}(A)$ .
-
(5)
$\operatorname {Aut}(\mathfrak {A})$ has finitely many closed non-isomorphic supergroups in
$\operatorname {Sym}(A)$ .
Proof The equivalence of items (1) and (2) is Proposition 6.36 in [Reference Bodirsky and Bodor11]. For the direction (1)
$\rightarrow $
(3) we only need to show that item (1) implies that every reduct of
$\mathfrak {A}$
is interdefinable with a structure with a finite relational signature. This follows for instance from Lemma 6.35 in [Reference Bodirsky and Bodor11].
In order to show the implication (3)
$\rightarrow $
(1) it is enough to show that
$\mathfrak {A}$
has finitely many reducts
$\mathfrak {B}$
with
$m(\mathfrak {B})\leq N$
up to interdefinability. Let R be a relation of
$\mathfrak {B}$
with arity at most N. Then R is interdefinable with the relation
$\{(x_1,\dots ,x_N)\colon (x_1,\dots ,x_{\operatorname {ar}(R)})\in R\}$
. Therefore we can assume without loss of generality that every relation of
$\mathfrak {B}$
is N-ary. Since
$\mathfrak {B}$
is a reduct of
$\mathfrak {A}$
it follows that every N-ary relation of
$\mathfrak {B}$
is a union of some orbits of
$\mathfrak {A}$
. Therefore there are at most
$2^{o_N(\mathfrak {A})}$
many possible choices for an N-ary relation of
$\mathfrak {B}$
, and hence there are at most
$2^{2^{o_N(\mathfrak {A})}}$
many choices for reducts
$\mathfrak {B}$
of
$\mathfrak {A}$
with
$m(\mathfrak {B})\leq N$
. Since
$\mathfrak {A}$
is
$\omega $
-categorical we know that
$o_N(\mathfrak {A})$
is finite, and hence
$2^{2^{o_N(\mathfrak {A})}}$
is also finite.
The equivalences (1)
$\leftrightarrow $
(4) and (2)
$\leftrightarrow $
(5) follow from Theorem 2.13 and Lemma 2.14.
Definition 2.18. We call a structure finitely homogenizable if it is interdefinable with a homogeneous structure with a finite relational signature.
We denote by
$\mathcal {FH}$
the class of finitely homogenizable structures.
Clearly the class
$\mathcal {FH}$
is closed under bidefinability, and by Fact 2.16 we know that every structure in
$\mathcal {FH}$
is
$\omega $
-categorical. Thus it follows from Lemma 2.14 that the structures in
$\mathcal {FH}$
can fully be described in terms of their automorphisms groups. We give a concrete such description below, as well as other equivalent conditions to check whether a structure is contained in the class
$\mathcal {FH}$
. We first need a couple of notations.
Notation 2.19. For a tuple
$\vec {a}\in X^n$
we denote by
$a_i$
the ith coordinate of
$\vec {a}$
. For a function
$\pi \colon k\rightarrow n$
we denote by
$\vec {a}_{\pi }$
the tuple
$(a_{\pi (0)},\dots ,a_{\pi (k-1)})$
.
Notation 2.20. Let
$\mathfrak {A}$
be a relational structure, and let
$m\in \omega $
. Then we denote by
$\Delta _m(\mathfrak {A})$
the structure whose domain is A, and whose relations are all subsets of
$A^m$
which are definable in
$\mathfrak {A}$
.
It is clear from the definition that if
$m'\leq m$
then every
$m'$
-ary relation of
$\mathfrak {A}$
is quantifier-free definable in
$\Delta _m(\mathfrak {A})$
, that is
$\Delta _{m'}(\mathfrak {A})$
is a quantifier-free reduct of
$\Delta _m(\mathfrak {A})$
.
If
$\mathfrak {A}$
is
$\omega $
-categorical then every type over
$\mathfrak {A}$
is principal, and thus in this case
$\Delta _m(\mathfrak {A})$
is a first-order reduct of
$\mathfrak {A}$
for all
$m\in \omega $
.
For a structure
$\mathfrak {A}$
and tuples
$\vec {u},\vec {v}\in A^n$
we write
$\vec {u}\approx _{\mathfrak {A}}\vec {v}$
iff
$\vec {u}$
and
$\vec {v}$
have the same type over
$\mathfrak {A}$
, and we write
$\vec {u}\not \approx _{\mathfrak {A}} \vec {v}$
iff
$\vec {u}\approx _{\mathfrak {A}} \vec {v}$
does not hold. In the case when
$\mathfrak {A}$
is
$\omega $
-categorical
$\vec {u}\approx _{\mathfrak {A}} \vec {v}$
holds if and only if
$\vec {u}$
and
$\vec {v}$
are in the same orbit of
$\operatorname {Aut}(\mathfrak {A})$
.
Lemma 2.21. Let
$\mathfrak {A}$
be an
$\omega $
-categorical structure, and let
$m\geq 2$
. Then the following are equivalent:
-
(1)
$\mathfrak {A}$ is interdefinable with a homogeneous relational signature structure
$\mathfrak {A}'$ with
$m(\mathfrak {A}')\leq m$ .
-
(2)
$\mathfrak {A}$ is interdefinable with a homogeneous relational signature structure all of whose relations have arity exactly m.
-
(3)
$\mathfrak {A}$ and
$\Delta _m(\mathfrak {A})$ are interdefinable, and
$\Delta _m(\mathfrak {A})$ is homogeneous.
-
(4) For all
$\vec {u},\vec {v}\in A^n$ if
$\vec {u}\not \approx _{\mathfrak {A}}\vec {v}$ then there exists a function
$\pi \colon m\rightarrow n$ such that
$\vec {u}_{\pi }\not \approx _{\mathfrak {A}}\vec {v}_{\pi }$ .
Proof The implication (2)
$\rightarrow $
(1) is obvious. The converse follows from the fact that if
$m'\leq m$
then every
$m'$
-ary relation of
$\mathfrak {A}'$
is quantifier-free definable from an m-ary relation of
$\mathfrak {A}'$
.
The implication (3)
$\rightarrow $
(2) is obvious. For the converse implication let us assume that item (2) holds, and let
$\mathfrak {A}'$
be a structure as it is dictated in the condition. Then
$\Delta _m(\mathfrak {A})=\Delta _m(\mathfrak {A}')$
. Since
$\mathfrak {A}$
, and thus also
$\mathfrak {A}'$
is
$\omega $
-categorical it follows that every definable relation of
$\mathfrak {A}'$
of arity m is a finite union of some m-orbits of
$\mathfrak {A}'$
. Therefore
$\mathfrak {A}'$
is quantifier-free definable in
$\Delta _m(\mathfrak {A}')$
. Since
$\Delta _m(\mathfrak {A}')$
is reduct of
$\mathfrak {A}'$
and
$\mathfrak {A}'$
is homogeneous it follows that
$\Delta _m(\mathfrak {A}')$
is also quantifier-free definable in
$\mathfrak {A}'$
. Therefore
$\Delta _m(\mathfrak {A}')$
is interdefinable with
$\mathfrak {A}$
, and it is homogeneous.
(3)
$\rightarrow $
(4). Let us assume that condition (3) holds. We can assume without loss of generality that
$\mathfrak {A}=\Delta _m(\mathfrak {A})$
. We show the contrapositive of the statement in item (4). Let us assume that for all
$\pi \colon m\rightarrow n$
we have
$\vec {u}_{\pi }\approx _{\mathfrak {A}}\vec {v}_{\pi }$
. Then by definition we have
$R(\vec {u}_{\pi })\Leftrightarrow R(\vec {v}_{\pi })$
for all
$\pi \colon m\rightarrow n$
and for all relations R of
$\mathfrak {A}$
with arity m. Since
$\mathfrak {A}=\Delta _m(\mathfrak {A})$
we know that the arity of every relation of
$\mathfrak {A}$
is exactly m. This means that the tuples
$\vec {u}$
and
$\vec {v}$
have the same atomic type over
$\mathfrak {A}$
. Since
$\mathfrak {A}$
is homogeneous, it has quantifier elimination, and hence
$\vec {u}$
and
$\vec {v}$
have the same type over
$\mathfrak {A}$
.
(4)
$\rightarrow $
(3). We show the contrapositive. Let us assume that item (3) does not hold. This means that
$\mathfrak {A}$
has a relation R which is not quantifier-free definable over
$\Delta _m(\mathfrak {A})$
. Then there exist tuples
$\vec {u}\in R, \vec {v}\in \bar {R}$
such that for every relation S of
$\Delta _m(\mathfrak {A})$
, and
$\pi \colon m\rightarrow n$
we have
$\vec {u}_{\pi }\in S$
if and only if
$\vec {v}_{\pi }\in S$
. By definition this implies that for all
$\pi \colon m\rightarrow n$
the tuples
$\vec {u}_{\pi }$
and
$\vec {v}_{\pi }$
have the same type over
$\mathfrak {A}$
. Thus the tuples
$\vec {u}$
and
$\vec {v}$
witness that Condition (4) fails.
The following is a direct consequence of Lemma 2.21.
Corollary 2.22. Let
$\mathfrak {A}$
be an
$\omega $
-categorical structure. Then the following are equivalent:
-
(1)
$\mathfrak {A}\in \mathcal {FH}$ .
-
(2)
$\mathfrak {A}$ is interdefinable with
$\Delta _m(\mathfrak {A})$ for some m.
-
(3) There exists an
$m\in \omega $ such that for all
$\vec {u},\vec {v}\in A^n$ if
$\vec {u}\not \approx _{\mathfrak {A}}\vec {v}$ then there exists a function
$\pi \colon m\rightarrow n$ such that
$\vec {u}_{\pi }\not \approx _{\mathfrak {A}}\vec {v}_{\pi }$ .
2.8. Finite boundedness
In the study of homogeneous structures with a finite relational signature we are particularly interested in the ones which are also finitely bounded.
Definition 2.23. Let
$\tau $
be a relational signature, and let
$\mathcal {C}$
be a set of finite
$\tau $
-structures. Then we denote by
$\operatorname {Forb}(\mathcal {C})$
the class of those finite
$\tau $
-structures which do not embed any structure from
$\mathcal {C}$
.
For a relational structure
$\mathfrak {A}$
we call the class of finite structures which embed into
$\mathfrak {A}$
the age of
$\mathfrak {A}$
, and we denote this class by
$\operatorname {Age}(\mathfrak {A})$
.
A relational structure
$\mathfrak {A}$
is called finitely bounded if its signature
$\tau :=\Sigma (\mathfrak {A})$
is finite, and there is finite set
$\mathcal {C}$
of
$\tau $
-structures such that
$\operatorname {Age}(\mathfrak {A})=\operatorname {Forb}(\mathcal {C})$
.
Next we give a couple of equivalent conditions for a homogeneous structure with a finite relational signature to be finitely bounded which we will need later.
Notation 2.24. Let
$\mathcal {C}$
be a class of finite
$\tau $
-structures. Then:
-
• We denote by
${\mathcal {C}}^{c}$ the class of those finite
$\tau $ -structures which are not in
$\mathcal {C}$ .
-
• We denote by
$\operatorname {Min}(\mathcal {C})$ the class of minimal structures in
$\mathcal {C}$ , i.e., the class of those structures in
$\mathcal {C}$ which do not have a proper substructure which is also in
$\mathcal {C}$ .
Lemma 2.25. Let
$\mathfrak {A}$
be a structure with a finite relational signature
$\tau $
. Then the following are equivalent:
-
(1)
$\mathfrak {A}$ is finitely bounded.
-
(2)
$\operatorname {Min}({\operatorname {Age}(\mathfrak {A})}^{c})$ contains finitely many structures up to isomorphism.
-
(3) There exists some m such that the size of every structure in
$\operatorname {Min}({\operatorname {Age}(\mathfrak {A})}^{c})$ is at most m.
Proof The equivalence of conditions (2) and (3) is clear since
$\tau $
is finite.
(2)
$\rightarrow $
(1). Let
$\mathcal {F}$
be a finite subset of
$\operatorname {Min}({\operatorname {Age}(\mathfrak {A})}^{c})$
so that every structure in
$\operatorname {Min}({\operatorname {Age}(\mathfrak {A})}^{c})$
is represented in
$\mathcal {F}$
. We claim that
$\operatorname {Age}(\mathfrak {A})=\operatorname {Forb}(\mathcal {F})$
. Let us suppose that
$\mathfrak {B}$
is a finite
$\tau $
-structure not contained in
$\operatorname {Forb}(\mathcal {F})$
. Then
$\mathfrak {B}$
has a substructure
$\mathfrak {B}'$
which is isomorphic to some structure in
$\mathcal {F}$
. In particular
$\mathfrak {B}'\not \in \operatorname {Age}(\mathfrak {A})$
. Since
$\operatorname {Age}(\mathfrak {A})$
is closed under taking substructures it follows that
$\mathfrak {B}\not \in \operatorname {Age}(\mathfrak {A})$
. For the other direction let us assume that
$\mathfrak {B}\not \in \operatorname {Age}(\mathfrak {A})$
. Then
$\mathfrak {B}$
contains a minimal structure with this property, say
$\mathfrak {B}'$
. Then by definition
$\mathfrak {B}'$
is isomorphic to some structure in
$\mathfrak {C}\in \mathcal {F}$
, and hence
$\mathfrak {C}$
embeds into
$\mathfrak {B}$
, that is
$\mathfrak {B}\not \in \operatorname {Forb}(\mathcal {F})$
.
(1)
$\rightarrow $
(3). Let us assume that
$\operatorname {Age}(\mathfrak {A})=\operatorname {Forb}(\mathcal {F})$
for some finite set
$\mathcal {F}$
of
$\tau $
-structures, and let m be the maximum of the sizes of structures in
$\mathcal {F}$
. Let
$\mathfrak {B}\in \operatorname {Min}({\operatorname {Age}(\mathfrak {A})}^{c})$
. Since
$\mathfrak {B}\not \in \operatorname {Age}(\mathfrak {A})=\operatorname {Forb}(\mathcal {F})$
it follows
$\mathfrak {B}$
has a substructure
$\mathfrak {B}'$
which is isomorphic to some
$\mathfrak {C}\in \mathcal {F}$
. But in this case
$\mathfrak {B}'$
is also not in
$\operatorname {Age}(\mathfrak {A})$
. Then the minimality of
$\mathfrak {B}$
implies that in fact
$\mathfrak {B}=\mathfrak {B}'$
, and hence
$|\mathfrak {B}|\leq m$
.
Notation 2.26. For a structure
$\mathfrak {A}$
we denote by
$b(\mathfrak {A})$
the maximum of
$m(\mathfrak {A})$
(see Notation 2.6) and the maximum of sizes of structures in
$\operatorname {Min}({\operatorname {Age}(\mathfrak {A})}^{c})$
.
$b(\mathfrak {A})$
is defined to be
$\infty $
if such a maximum does not exist.
Lemma 2.25 shows that if
$\mathfrak {A}$
is homogeneous then it is finitely bounded if and only if
$b(\mathfrak {A})$
is finite.
Notation 2.27. We denote by
$\mathcal {FBH}$
the class of those structures which are interdefinable with a finitely bounded homogeneous relational structure.
We can argue the same way as in the previous section that the structures in
$\mathcal {FBH}$
can be described in terms of their automorphisms. In this case, however, we do not know of any nice such description. Nevertheless we know that the class
$\{\operatorname {Aut}(\mathfrak {A})\colon \mathfrak {A}\in \mathcal {FBH}\}$
is closed under taking finite direct products, infinite copies, and rather surprisingly (as we will see in Section 4) it is also closed under taking finite index supergroups.
2.9. Unions and copies of structures
Let us consider the following two constructions on structures.
Definition 2.28. Let
$\mathfrak {A}_1,\dots ,\mathfrak {A}_n$
be relational structures with pairwise disjoint domains.
Then we define the disjoint union
$\mathfrak {B}$
of the structures
$\mathfrak {A}_1,\dots ,\mathfrak {A}_n$
, denoted by
$\biguplus _{i=1}^n\mathfrak {A}_i$
such that:
-
• the domain set of
$\mathfrak {B}$ is
$\bigcup _{i=1}^nA_i$ ,
-
• its signature is
$\bigcup _{i=1}^n\Sigma (\mathfrak {A}_i)$ together with some pairwise different unary symbols
$U_1,\dots ,U_n\not \in \bigcup _{i=1}^n\Sigma (\mathfrak {A}_i)$ ,
-
•
$U_i^{\mathfrak {B}}=A_i$ for all
$1\leq i\leq n$ , and
-
•
$R^{\mathfrak {B}}=R^{\mathfrak {A}_i}$ for all
$R\in \Sigma (\mathfrak {A}_i)$ .
Definition 2.29. Let
$\mathfrak {A}$
be a relational structure. Then we define the structure
$\mathfrak {A}\wr \omega $
, called infinitely many copies of
$\mathfrak {A}$
, as follows.
-
• The domain set of
$\mathfrak {A}\wr \omega $ is
$A\times \omega $ .
-
• Its signature is
$\Sigma (\mathfrak {A})\cup \{E\}$ where E is a binary relational symbol not contained in
$\Sigma (\mathfrak {A})$ .
-
•
$E^{\mathfrak {A}\wr \omega }=\{((a_1,n),(a_2,n))\colon a_1,a_2\in A, n\in \omega \}$ .
-
•
$R^{\mathfrak {A}\wr \omega }=\{((a_1,n),\dots ,(a_k,n))\colon (a_1,\dots ,a_k)\in R^{\mathfrak {A}}, n\in \omega \}$ for all
$R\in \Sigma (\mathfrak {A})$ of arity k.
The following two propositions are easy consequences of the definitions above.
Proposition 2.30. Let
$\mathfrak {A}_1,\dots ,\mathfrak {A}_n$
be relational structures as in Definition 2.28. Then
$\operatorname {Aut}(\biguplus _{i=1}^n\mathfrak {A}_i)=\prod _{i=1}^n\operatorname {Aut}(\mathfrak {A}_i)$
.
Proposition 2.31. Let
$\mathfrak {A}$
be a relational structure. Then
$\operatorname {Aut}(\mathfrak {A}\wr \omega )=\operatorname {Aut}(\mathfrak {A})\wr \operatorname {Sym}(\omega )$
.
The following two lemmas play a crucial role in our proofs later.
Lemma 2.32. Let
$\mathfrak {A}_1,\dots ,\mathfrak {A}_n$
be structures with pairwise disjoint domains, and let
$\mathfrak {A}:=\prod _{i=1}^n\mathfrak {A}_i$
. Then:
-
(1) if
$\mathfrak {A}_1,\dots ,\mathfrak {A}_n$ are all homogeneous, then so is
$\mathfrak {A}$ , and
-
(2) if
$\mathfrak {A}_1,\dots ,\mathfrak {A}_n$ are all finitely bounded, then so is
$\mathfrak {A}$ .
Lemma 2.33. Let
$\mathfrak {A}$
be homogeneous structures. Then
$\mathfrak {A}\wr \omega $
is homogeneous, and
$\mathfrak {A}$
is finitely bounded, then so is
$\mathfrak {A}\wr \omega $
.
Proof of Lemma 2.32
Item (1) follows from a straightforward induction using Lemma 3.1 in [Reference Bodirsky8].
As for item (2) let us assume that
$\operatorname {Age}(\mathfrak {A})=\operatorname {Forb}(\mathcal {F}_i)$
for some finite sets of structures
$\mathcal {F}_i$
for
$1\leq i\leq n$
. For a
$\Sigma (\mathfrak {A}_i)$
-structure
$\mathfrak {B}$
let
$(\mathfrak {B};U_i)$
denote the structure
$\mathfrak {B}$
expanded by a unary relation denoted by
$U_i$
which contains all elements of
$B_i$
. Let
$\mathcal {G}$
be the set of one-element
$\{U_1,\dots ,U_n\}$
-structures
$\mathfrak {B}$
such that either
$\mathfrak {B}^{U_i}=\emptyset $
for all i or
$\mathfrak {B}^{U_i}=\mathfrak {B}^{U_j}=B$
for some
$i\neq j$
. Then it follows from the construction that

Therefore
$\mathfrak {A}$
is finitely bounded.
Proof of Lemma 2.33
Let us assume that the tuples

have the same quantifier-free type over
$\mathfrak {A}\wr \omega $
. We have to show that
$\vec {u}$
and
$\vec {v}$
are in the same orbit of
$\operatorname {Aut}(\mathfrak {A}\wr \omega )=\operatorname {Aut}(\mathfrak {A})\wr \operatorname {Sym}(\omega )$
. For an
$n\in \omega $
let us denote by
$\vec {u}_n$
(
$\vec {v}_n$
) the subtuple of
$\vec {u}$
(
$\vec {v}$
) containing those elements whose second coordinate is n. By definition it follows that
$m_i=m_j$
iff
$n_i=n_j$
for all
$1\leq i,j\leq k$
. Thus there exists a permutation
$\sigma \in \operatorname {Sym}(\omega )$
such that for
$\vec {u}_m:=((a_{i_1},m),\dots ,a_{i_k},m))$
we have
$n_{i_1}=\dots =n_{i_k}=\sigma (m)$
. Moreover by the definition of
$\mathfrak {A}\wr \omega $
it follows that the tuples
$(a_{i_1},\dots ,a_{i_k})$
and
$(b_{i_1},\dots ,b_{i_k})$
have the same quantifier-free type in
$\mathfrak {A}$
. Thus by the homogeneity of
$\mathfrak {A}$
it follows that there exists some
$\alpha _m\in \operatorname {Aut}(\mathfrak {A})$
such that
$\alpha _m(a_{i_j})=b_{i_j}$
for all
$j=1,\dots ,k$
. Now let
$\gamma \in \operatorname {Sym}(A\times \omega )$
be defined as
. Then
$\gamma \in \operatorname {Aut}(\mathfrak {A})\wr \operatorname {Sym}(\omega )$
and by our construction
$\gamma (a_i,m_i)=(\alpha _{m_i}(a_i),\beta (m_i))=(b_i,n_i)$
for all
$1\leq i\leq k$
.
For the second part of the lemma let us assume that
$\operatorname {Age}(\mathfrak {A})=\operatorname {Forb}(\mathcal {F})$
for some finite set of structures
$\mathcal {F}$
. Let
$\mathcal {G}_0$
be a finite set of structures in the signature
$\{E\}$
so that
$\mathfrak {A}\in \operatorname {Forb}(\mathcal {G}_0)$
if and only if
$E^{\mathfrak {A}}$
is an equivalence relation on
$\mathfrak {A}$
. We can do this by listing all three-element
$\{E\}$
-structures where E does not define an equivalence relation. For a
$\Sigma (\mathfrak {A})$
-structure
$\mathfrak {C}$
let
$\mathfrak {C}^*$
denote the structure
$\mathfrak {C}$
expanded by a binary relation, denoted by E which contains every pair in
$C^2$
. Let

Then it is straightforward to check that
$\operatorname {Forb}(\mathcal {G})$
is exactly the age of
$\mathfrak {A}\wr \omega $
. Therefore
$\mathfrak {A}\wr \omega $
is finitely bounded.
Corollary 2.34. The classes
$\mathcal {FH}$
and
$\mathcal {FBH}$
are closed under taking finite disjoint unions and infinite copies.
3. Hereditarily cellular structures
Definition 3.1. Let T be a complete theory in some language L. Then we say that T is monadically stable if every expansion of T by unary predicates is stable. A first-order structure
$\mathfrak {A}$
is called monadically stable if its first-order theory is monadically stable.
We denote the class of countable
$\omega $
-categorical monadically stable structures by
$\mathcal {M}$
. For a natural number k we denote by
$\mathcal {M}_k$
the class of structures in
$\mathcal {M}$
whose Morley rank is at most k.
Lachlan showed in [Reference Lachlan34] that the structures in
$\mathcal {M}$
can be described in terms of their automorphism group. For the description of these groups we first need a couple of definitions.
Definition 3.2. Let
$G\leq \operatorname {Sym}(X)$
be a permutation group. Then a triple
$P=(K, \nabla , \Delta )$
is called an
$\omega $
-partition of G iff:
-
(1)
$K\subset X$ is finite, and it is fixed setwise by G.
-
(2)
$\Delta $ and
$\nabla $ are congruences of
$G|_{X\setminus K}$ with
$\Delta \subset \nabla $ .
-
(3)
$\nabla $ has finitely many classes.
-
(4) Every
$\nabla $ -class is a union of