Automorphisms and strongly invariant relations

We investigate characterizations of the Galois connection Aut\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Aut}\,}}$$\end{document}-sInv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{sInv}\,}}$$\end{document} between sets of finitary relations on a base set A and their automorphisms. In particular, for A=ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=\omega _1$$\end{document}, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under sInv Aut\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{sInv Aut}}$$\end{document}. Our structure (A, R) has an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.


Introduction
Our main question is easy to formulate.Let R be a set of finitary relations on a nonempty base set A, and let Aut R denote the set of all automorphisms of the structure (A ; (̺) ̺∈R ).Conversely, if G is a set of permutations on A, then sInv G denotes the the set of all relations σ on A such that all permutations in G are automorphisms of σ. (Formal Definitions follow in the next section.)The question is: How can we characterize the relation sets of the form sInv Aut R ?
Of course, the operator sInv Aut is a closure operator, and the operator pair sInv-Aut forms a Galois connection between sets of relations on A and sets of permutations on A. We can reformulate our problem as "Which sets R of relations are Galois-closed, i.e., satisfy R = sInv Aut R?" or: "Describe the closure operator sInv Aut internally", i.e., without explicit reference to permutations.
Probably the first one who investigated this question in a systematic way was Marc Krasner.Influenced by the Galois connection between permutation groups and field extensions he tried to 'generalize the notion of a field' [7].Instead of the action of permutations on field elements, he considered the more complex action on relations.For finite base sets A he described the closed sets of relations with the help of some operations on relations.A logical operation on relations is an operation, definable by a formula of the first order logic.(For details see the next section.)We call a set of relations a Krasner algebra if it is closed under all logical operations.For finite A, the Galois closed sets of relations are exactly the Krasner algebras.(At this point we remark that our notation differs from Krasner's original notation.) It is easy to extend this characterization to countable base sets A: In this case the Galois closed relation sets are exactly those Krasner algebras that are additionally closed under arbitrary intersections ( -closed Krasner algebras).
But this is no longer true for the general case of uncountable sets A. For this case there exists a characterization by R. Pöschel [12,13] with the help of additional operations of uncountable arity.But the use of such operations is not very satisfying.Therefore we continue to look for better results.
One reason for the existence of -closed Krasner algebras that are not Galois closed is the fact that first order logic is simply "too weak" to distinguish between sets of different infinite cardinalities.Consequently, it is a natural idea to replace the logical operations by a stronger class of operations.An n-ary operation F on relations is called invariant, if the following identity holds for all permutations g and all relations ̺ 1 , ̺ n (with appropriate arities) on A: Date: 2003-09-09.The second author is grateful to the department of mathematics at Rutgers university for their hospitality during a visit in Fall 2002.
The third author is supported by NSF and the Israel Science Foundation.Publication 822.
Clearly, every Galois closed set of relations is -closed and closed under all invariant operations.But it was unknown whether the converse is also true.The problem is: Does there exist a set of relations that is -closed and closed under all invariant operations, but not Galois closed for sInv-Aut?([3, Problem 2.5.2].) Surprisingly, the answer to this question is yes!In the main part of our article, section 3, we give a model theoretical construction of such a set of relations on a base set A of cardinality ω 1 .
Finally, in section 4, we give a characterization of the Galois closed relation sets with the help of additional invariant infinitary operations.In contrast to Pöschels characterization, we restrict these infinite arities to be countable.Section 3 shows that we cannot restrict the arities to be finite, so this seems to be the best possible result.

Preliminaries.
Notation.Throughout, let A denote a nonempty base set.Write ω for the set of all natural numbers (and at the same time the first infinite ordinal).An m-ary relation on A is a subset of A m , the set of all m-ary relations is denoted by Rel (m) (A), and Rel(A) := 1 m∈ω Rel (m) (A) is the set of all finitary relations.If R ⊆ Rel(A), then R (m) := R ∩Rel (m) (A) .We do not distinguish between relations and predicates, therefore a ∈ ̺ and ̺(a) have the same meaning.The set of all permutations on A is denoted by Sym(A).For g ∈ Sym(A) and a = (a 1 , . . ., a m ) ∈ A m we put g(a) := (g(a 1 ), . . ., g(a m )), and for ̺ ⊆ A m we write g[̺] := {g(a) | a ∈ ̺}.Let g ∈ Sym(A) and ̺ ∈ Rel(A).We say that g is an automorphism of ̺, or that g strongly preserves ̺, or that ̺ is a strongly invariant relation for g, if For R ⊆ Rel(A) and G ⊆ Sym(A) we define operators Aut : PRel(A) → P Sym(A) and sInv : P Sym(A) → PRel(A): = ̺ for all g ∈ G} (For a set X, PX denotes the set of all subsets of X.) The operator pair sInv-Aut forms a Galois connection between sets of permutations and sets of relations on A, i.e. the following conditions are satisfied: • Consequently, the operators sInv Aut : PRel(A) → PRel(A) and Aut sInv : P Sym(A) → P Sym(A) are closure operators.The sets of relations and the sets of permutations which are closed under these closure operators are called Galois closed (with respect to the Galois connection sInv-Aut).Characterizing a Galois connection means to describe the Galois closed sets without referring to the connection itself.
In our article we want to find and discuss characterizations of our Galois connection sInv-Aut.There exist many similar Galois connections between sets of relations and sets of different kinds of functions, and they turned out to be useful especially for the investigation of finite mathematical structures.As a general source, we refer to [11] and the list of references given there.Here we are interested in characterizations for infinite base sets A.
The main tool for the description of the closed sets of relations are operations on relations.These operations are of the form Special operations on relations are the logical operations which can be defined with the help of first order formulas.More exactly: Let ϕ(P 1 , . . ., P n ; x 1 , . . ., x m ) be a formula with predicate symbols P i (of arity m i ), where all free variables are in {x j | 1 j m}.We define where ϕ A (̺ 1 , . . ., ̺ n , a 1 , . . ., a m ) means that ϕ holds in the structure A ; ̺ 1 , . . ., ̺ n for the evaluation x j := a j , (1 j m).
Properties of Galois closed relation sets.
Definition 2.1.A set R ⊆ Rel(A) is called a Krasner algebra (KA) on A, if R is closed under all logical operations. 1f Q ⊆ Rel(A), then Q KA denotes the Krasner algebra generated by Q, i.e. the least set of relations on A that contains Q and is closed under all logical operations.
A set (2) R is closed under arbitrary intersections i.e. for all m and all Q ⊆ R (m) we have Q ∈ R (m) .(Here we put ∅ = A m .) If Q ⊆ Rel(A), then Q KA, denotes the least -closed Krasner algebra containing Q.
Clearly, KA and KA, are closure operators with Q KA ⊆ Q KA, for all Q ⊆ Rel(A).If a set R of relations is -closed and closed under complementation C (e.g. if R = R KA, ), then it is also closed under arbitrary unions.
The next Lemma gives the obvious connection between these notions and the Galois closed relation sets.For a proof we refer e.g. to [3].
(Galois closed sets of relations are sometimes called Krasner clones.So every Krasner clone is a Krasner algebra, but not vice versa.) and a ∈ A m .Then the following hold.
(1) Proof.We have Partial automorphisms.The last Lemma can be used to find characterizations in some special cases.
Definition 2.6.A partial automorphism f of a relation set Q ⊆ Rel(A) (or of the structure such that for all σ ∈ Q, m = arity(σ) and all a 1 , . . ., a m ∈ dom f , we have: ) is said to be homogeneous, if every finite partial automorphism can be extended to an automorphism of Q.
Let a = (a 1 , . . ., a m ) and b = (b 1 , . . ., b m ).We will use 2.5.So, assuming a ∼ Q b, we have to find an automorphism g with g(a Because of the homogeneity of Q, f can be extended to an automorphism g ∈ Aut Q. Therefore b = g(a) for some g ∈ Aut Q.
Relation sets of the form sInv Aut Q are homogeneous.Hence, a -closed Krasner algebra is Galois closed if and only if it is homogeneous.
The Galois closed permutation sets.We want to have a short look at the other side of our Galois connection.The characterization of the Galois closed permutation sets is well known ( [5]) and provides no difficulties.We need an additional closure operator Loc o : P Sym(A) → P Sym(A): For the proof we refer to [5].The operator Loc o is a topological closure operator, multiplication and inversion of permutations are continuous with respect to the underlying topology.Therefore, the Galois closed automorphism sets are characterized as certain topological groups.For a more detailed discussion we refer to [4, 4.1].
A first characterization of the Galois closed relation sets.In [12] and [13], R. Pöschel characterized the closed sets of relations with the help of infinitary operations.Let I be an arbitrary index set, let m, m i ∈ ω \ {0} (i ∈ I).For an I-tuple

Therefore every automorphism of {̺
Consequently, every Galois closed set of relations is closed under strong superposition.
If R is closed under strong superposition, then we can choose I and (b i ) i∈I such that all finite sequences with elements of A occur among the b i .(This is possible with This implies for every b ∈ Γ R (a) the existence of a permutation g ∈ Sym(A) with b = g(a) and with g(c) ∈ Γ R (c) for all n and all c ∈ A n .This g is an automorphism of R, therefore 2.5 implies that R is Galois closed.
As seen in the proof, we can restrict the arities of the strong superpositions to |I| = |A|.Nevertheless, to be closed under strong superposition is a very strong condition.It immediately implies the existence of the necessary automorphisms in the sense of 2.5.Therefore, we continue to find better characterizations.
A Characterization for countable base set A. For finite base set A, the Galois closed relation sets are exactly the Krasner algebras ( [7,8,9,11]).This result can be extended to the countable case: Theorem 2.10.Let A be a countable or finite set and Proof.For the proof we refer to [2, 3.3.6.(v)] or to [3, 2.4.4.(i)].One direction is provided by Lemma 2.2.For the other direction we can use a back & forth construction to obtain the automorphisms that are necessary to apply Lemma 2.5.
The next example shows that this characterization cannot be extended to uncountable sets.
( In each of these models M κ , the set R of first order definable relations (without parameters) is clearly a Krasner algebra and is trivially closed under (since, by Ryll-Nardzewski's theorem, for any k there are only finitely many k-ary relations in R).
But in each model M κ the set ̺ * is a (higher order) definable subset of M κ , hence This shows that R is not Galois closed.
Invariant operations.In the last example, the logical operations, together with arbitrary intersections, are too weak to provide the closure under sInv Aut.In particular, with logical operations it is not possible to distinguish between sets of different infinite cardinality.So the next idea to obtain a characterization is to replace the logical operations by a family of stronger operations on relations.
Definition 2.12.An operation if for all g ∈ Sym(A) and all ̺ i ∈ Rel mi (A) the following identity holds: If Q ⊆ Rel(A), then Q inv denotes the closure of Q under all invariant operations, and Q inv, denotes the least set of relations that is closed under all invariant operations, -closed and contains the set Q.
(The notations "logical operations" and "invariant operations" are adopted from [6].)The operators Q → Q inv and Q → Q inv , are closure operators, and we have We collect some easy properties of the invariant operations.
(1) Every logical operation is invariant.If A is finite, then every invariant operation is logical.
(2) The invariant operations form a clone, i.e. the superposition of invariant operations is again an invariant operation.
and closed under all invariant operations, R = R inv , .
The next Lemma shows that invariant operations are sufficient, if we have only finitely many relations.
It is easy to verify that F has the desired properties.
Let H : PZ → PZ be a closure operator on a set Z. The algebraic part of H is the closure operator H is algebraic if H = H alg .Lemma 2.14 shows that the closure operator Q → Q inv is the algebraic part of sInv Aut.More exactly, for all Q ⊆ Rel(A) we have For finite base set A all these closure operators coincide.For countable A, the operators KA , inv and All these properties now lead to the conjecture, that the Galois closed sets of relations are exactly the sets of relations that are -closed and closed under all invariant operations.In order to verify this conjecture, we have to answer one question: Does there exist a set R of relations that is -closed and closed under all invariant relations, but is not Galois closed?
This question was formulated as an open problem e.g. in [3, Problem 2.5.2].Surprisingly, the question has a positive answer and therefore the conjecture above is false.In the next section we will give a model theoretic construction of a relation set R with the mentioned properties.

A model theoretic construction
In this section we consider relational models of the form M = (M ; (̺ m ) 1 m∈ω ), where ̺ m ∈ Rel (m) (A) for all m.So our language L has exactly one relation symbol for every arity m. (We use the predicate symbols also to denote the corresponding relations.We assume that the following hold: (1) The theory Th(A) is ω-categorical (i.e., has up to isomorphism exactly one countable model).
(2) For all m, the reduct A [m] is homogeneous in the sense of 2.6.
(3) A is rigid, i.e.Aut A = {id A }.Then, letting R := ̺ 1 , ̺ 2 , . . .KA be the set of first order definable relations in A, we have: First note that R (as well as ̺ 1 , ̺ 2 , . . .̺ m KA ) is closed under arbitrary intersections, since (by Ryll-Nardzewski's theorem) there are only finitely many k-ary relations in R, for any k.
We now show that R is also closed under all invariant operations.We have . ., ̺ m }, but by (2) and 2.7 we have As both operators KA and inv are algebraic, this yields Clearly R is countable.But Aut R = {id A } and so sInv Aut R = Rel(A), which is an uncountable set.Consequently R inv , = sInv Aut R.
A clause in x 0 , . . ., x n is a conjunction K of literals in x 0 , . . ., x n , such that no literal will appear twice, and no literal appears in negated and unnegated form.
Please note that there are only finitely many clauses in x 0 , . . ., x n .The variable x 0 plays a special role -it has to appear in every literal.Now we formulate our theory T : Definition 3.3.T consists of (the universal closures of) the following formulas: Firstly, for all 1 m ∈ ω we have: Secondly, for all n ∈ ω and all clauses K = K(x 0 , . . ., x n ) in x 0 , . . ., x n we take the formula: (T2) 1 i<j n x i = x j → (∃x 0 ) K(x 0 , . . ., x n ) Informal Discussion 3.4.We will see below that the theory T is complete and ω-categorical.
Our aim is to construct an uncountable model M of this theory on the base set ω 1 in which the well-order (ω 1 , <) is definable (by a formula in higher order logic).To help us achieve this aim, we use the following "recommendation": this is just a recommendation, not a law.In order to also get homogeneity of the restricted models M [m] , we allow our model to disobey this recommendation, if there is a good reason for it.A good reason can be the desire to satisfy an axiom of our theory, or to extend a partial automorphism.
To keep track of the cases where the recommendation is not followed, we construct an auxiliary function h : M → ω, and we will demand the following "law", which is a relaxed version of the "recommendation": For all sufficiently long tuples (x 1 , . . ., x n ): and must not hold otherwise Here, "sufficiently long" is defined as: n > max(h(x 1 ), . . ., h(x n )).
whenever we violate our recommendation at a tuple (x 1 , . . ., x n ), we will define a sufficiently large value of h at one of the points x 1 , . . ., x n .
Before we investigate the theory T , we want to examine some technical definitions and lemmas.ω 1 denotes the first uncountable ordinal, ω 1 = {α | α < ω 1 }.ω 1 is well-ordered by <.The universes of all our models will be subsets of ω 1 .(2) for all n-tuples (a 1 , a 2 , . . ., a n ) ∈ M n that are weak for h the following condition holds: (1) The relation ❁ is transitive.
and M ω is the directed union of this chain (i.e., M ω = n∈ω M n , and each M n is also a submodel of M ω ), then M j ❁ M ω for all j < ω. (3) Similarly, if (M i ) i≤α (where α is a limit ordinal) is a continuous chain of models (i.e., M i ≤ M j for all i ≤ j ≤ α, and for each limit δ ≤ α we have M δ = i<δ M i ), and Proof. ( ) is weak for h, then either a ∈ (M n 2 \M n 1 ) or one of the a i belongs to M 3 \ M 2 .In the first case, max h 2 (a) = max h(a) < n, and a is weak for h 2 .Therefore (M 2 is a submodel of M 3 ), n (a) ⇐⇒ ̺ 2 n (a) ⇐⇒ a 1 < a 2 < . . .< a n .In the second case, max h 3 (a) max h(a) < n, therefore a is weak for h 3 and ̺ 3 n (a) ⇐⇒ a 1 < . . .< a n .
The next technical Lemma provides the basic step in our construction.
) be a countable model of our language L, such that M 0 ⊆ ω 1 and ̺ m (x 1 , . . ., x m ) → 1 i<j m x i = x j holds for all m.Moreover, let π 0 : M 0 •→ M 0 be a partial (finite or infinite) automorphism of the reduct M 0 [s] for some s ∈ ω and let α ∈ ω.Then there exists a countable model [s] such that the following hold: (1) M ω is a model of the theory T .
(3) π ω extends π 0 Proof.We construct M ω as the union of a chain of countable models M j with j ∈ ω.We will have M j ❁ M j+1 for all j ∈ ω, and for every j we will have a partial automorphism π j of M j [s] such that π j+1 extends π j , and dom(π j+1 ) ∩ im(π j+1 ) ⊇ M j .We explain the step from M j to M j+1 .Let M j = {a i | i ∈ ω} be an enumeration of the elements of M j .(The elements a i are not necessarily in the order, given by < in ω 1 .)Let K be the following set: This set is countable.Let (n l , a l , K l ) l∈ω be an enumeration of this set.Let B ⊆ ω 1 \ M j be a countable set of ordinals, and let B = {b k | k ∈ ω} be an enumeration of B. (Again, this enumeration need not necessarily follow the well-order < on ω 1 .)We put M j+1 := M j ∪B, and we have to define the relations ̺ j+1 m and the partial function π j+1 .Moreover, we must define a function h : B → ω, in order to establish the relation For all m and all m-tuples a ∈ M m j we define: ̺ j+1 m (a) : ⇐⇒ ̺ j m (a) This makes sure that M j M j+1 .Moreover, we define ¬̺ j+1 m (c 1 , . . ., c m ) for all c 1 , . . ., c m ∈ M j+1 with |{c 1 , . . ., c m }| < m.
For every k ∈ ω we will conduct a special task, where we define the value h(b k ), define a partial function p k : M j+1 •→ M j+1 such that p k+1 always extends p k .We start with p 0 := π j , and finally we will have π j+1 := k p k .Moreover, in every step we define the truth values of ̺ j+1 m (a) for some tuples a.
We distinguish three cases of steps k, depending on whether k ≡ 0, 1, or 2 mod (3).[A main point will be that the definitions in the various cases do not contradict each other.] Step k for k = 3l: If l = 0, then let p k := π j , otherwise p k := p k−1 remains unchanged.Let (n l , a l , K l ) be the element of K with index l.We define h(b k ) := n l + 1.Now, for every literal which occurs in K, we define the truth values in such a way, that K l (b k , a l ) becomes true.Thus, letting a l = (a l (1), . . ., a l (n l )), we define truth values for certain tuples from the set {a l (1), . . ., a l (n l ), b k } <ω \ {a l (1), . . ., a l (n l )}) <ω .
The largest index m of a literal which occurs in K l is n l + 1. Therefore all these tuples c satisfy max h(c) ≥ h(b k ) ≥ m and are strong for h.[Note that in no previous step have we committed ourselves to the truth value of ̺ j (c) for any tupel c in which b k appears.] Step k for k = 3l + 1: In this case we define h(b k ) := s, and we extend the partial function p k−1 .If dom p k−1 ⊇ M j , then we simply put p k := p k−1 and we are done.
Then, for all m s and all c 1 , . . ., c m ∈ im p k such that c 1 , . . ., c m are pairwise distinct and b k ∈ {c 1 , . . ., c m }, we define the truth value of is still not fixed (in particular, then the p −1 k (c i ) cannot all be in M j ), then we define both values, namely we put Because of the definition of h(b k ), the (c 1 , . . ., c m ) are always strong for h and hence are exempted from our "recommendation" 3.4.The other tuples, p −1 k (c), follow our recommendation anyway, whether or not they are h-strong.
[As before, note that tuples c in which b k appears have never been considered in any previous step k ′ < k.] Step k for k = 3l + 2: This step is similar to the previous step, but this time we take care of im p k rather than dom p k , or in other words: we reverse the roles of p k and p −1 k .We leave the details to the reader.By induction we obtain from these steps a model, where the relations ̺ j+1 m are only partially defined.In order to finish the definition, we put whenever the truth value of ̺ j+1 m (c) has not been defined during the inductive construction.From the construction, it is now clear that M j ❁ M j+1 .Moreover, π j+1 := k∈ω p k is a partial automorphism of M j+1 [s] that extends π i and satisfies M j ⊆ dom π j+1 and M j ⊆ im π j+1 .
Moreover, all formulas in T of the form 3.3(T2) are satisfied, whenever x 1 , . . ., x m ∈ M j .The countable set B ⊂ ω 1 \ M j was arbitrary.So, if the ordinal α is not in M 0 , then we can assume that α ∈ B, e.g. for j = 0. Consequently we will have α ∈ M ω in the end.
We form the directed union M ω := j∈ω M j Because of 3.6(2), we have M 0 ❁ M ω .Moreover, the union π ω := j∈ω π j is a bijective partial automorphism of M ω [s] , which is everywhere defined and surjective, i.e. it is an automorphism of M ω [s] which extends π 0 .Finally, if ψ(x 1 , . . ., x n ) is a formula of T and a 1 , . . ., a n ∈ M ω , then there exists j with a 1 , . . ., a n ∈ M j .Consequently ψ(a 1 , . . ., a n ) holds in M j+1 and all other extensions of M j , in particular it is true in M ω .Consequently M ω is a model of T .This finishes the proof.Now we collect some properties of our theory T .Lemma 3.8.
(1) T is consistent and has no finite models.(2) T has the property of elimination of quantifiers.
(5) If M , N are models of T and N is a submodel of M , N M , then N is an elementary submodel of M , i.e. for every formula ϕ(x 1 , . . ., x n ) and every a ∈ N n holds ϕ(a) in N iff it holds in M . Proof.
(1) Easy.The consistency of T is a corollary of 3.7, and the nonexistence of finite models is ensured by the formulas 3.3(T2).
For any equivalence relation θ on {1, . . ., n} we put: Let E n denote the set of all equivalence relations on {1, . . ., n}.Then Therefore also It is sufficient to show that every formula (∃x 0 )(K ∧ µ θ ) is equivalent to a quantifier free formula.If θ is not the equality relation, then we can replace any variable x j (j ≥ 1) by x i , where i is a representative of the equivalence class of j.If then a variable appears twice in an literal of K, then either the clause becomes false modulo T (if the literal is unnegated), or the literal can be omitted modulo T (if it is negated).In the end we obtain either formulas which are true (modulo T ) or false (modulo T ) or equivalent to a formula of the form (∃x 0 )(K ∧ T contains the formula 1 i<j n x i = x j → (∃x 0 )K, therefore (3) By (2), every closed formula is (mod T ) equivalent to true or false.
(4) Modulo T , there are only finitely many quantifier-free formulas in the variables x 1 , . . ., x n , namely, Boolean combinations of atomic formulas ̺ m (x i1 , . . ., x im ), for i 1 , . . ., i m ∈ {1, . . ., n} and m ≤ n. (Note that formulas ̺ m (x i1 , . . ., x im ) for m > n and i 1 , . . ., i m ≤ n are automatically false mod T , because of 3.3(T1).This implies ω-categoricity, by Ryll-Nardzewski's theorem.[Actually, we do not need ωcategoricity itself for our construction, we only need the fact that there are only finitely many first order definable k-ary relations, for any k.] (5) This is a consequence from the fact that T has elimination of quantifiers.
The results in 3.8 make sure that the condition 3.1(1) is satisfied for every model of T .Now we construct a model A, such that also the conditions 3.1(2) and (3) are satisfied.
We will obtain A as a directed union over an uncountable chain of models, A := i∈ω1 M i , such that every M i is a model of T , M i ❁ M i+1 and i ∈ M i+1 ⊂ ω 1 for all i ∈ ω 1 .Because of 3.8(4), this is an elementary chain, therefore A is again a model of T .Because of i ∈ M i+1 for all i ∈ ω 1 and M In order to obtain a model with homogeneous s-reducts, we have to make sure that certain partial automorphisms can be extended to automorphisms.For this reason, we use a triply-indexed family (π n,i,j ) n∈ω,i,j∈ω1,i j of partial automorphisms.
First we explain, what the π n,i,i are.If M is a countable model, then there are only countable many pairs (p, s) with s ∈ ω and p a finite partial automorphism of the s-reduct M [s] .Therefore there exists an enumeration (p n , s n ) n∈ω of all these finite partial automorphisms with corresponding s.Now, for i ∈ ω and M = M i we put π n,i,i := p n , and s n,i := s n .Therefore ( * * ) (π n,i,i ) n∈ω is a list of all finite partial automorphisms of all possible reducts M i [s] .
The π n,i,j with i < j will be extensions of π n,i,i .Now we explain how to construct the models M j and sequences of partial automorphisms (π n,i,j : i ≤ j) by transfinite induction on j ∈ ω 1 ).This construction will use the usual 'bookkeepingargument' to take care of ω 1 × ω many tasks in ω 1 steps.Let ω 1 = n∈ω,i∈ω1 C n,i be a partition of ω 1 into pairwise disjoint sets C n,i , such that |C n,i | = ω 1 for all (n, i) and min C n,i ≥ i.
If j = 0, then let M 0 be a countable model of T with M 0 ⊆ ω 1 .(The existence of such a model is clear from 3.7.)The π n,0,0 are defined as in ( * * ).
If j is a limit ordinal, then put M j := i<j M i .(As a directed union of an elementary chain of models of T , this is again a model of T .)The π n,j,j are defined as in ( * * ), and π n,i,j := i l<j π n,i,l .For a successor ordinal j + 1 we use Lemma 3.7: (1) We define M j+1 as follows.Let (n, i) be the pair with j ∈ C n,i .According to the Lemma, there exists a model M j+1 with j ∈ M j+1 ⊂ ω 1 and M j ❁ M j+1 , and there exists an extension of π n,i,j to a partial automorphism π of M j+1 [sn,i] with M j ⊆ dom π and M j ⊆ im π.
It is easy to verify by transfinite induction, that the π (n,i,j) are always partial automorphisms of M j [sn,i] , and that π n,i,j extends π n,i,k for all k with i k < j.
As mentioned above, we put A := i∈ω1 M i .
Lemma 3.9.If s ∈ ω, then every finite partial automorphism π of A [s] can be extended to an automorphism of A [s] .
Proof.π is finite, therefore there exists i ∈ ω with dom π ∪ im π ⊆ M i .M i is a submodel of A, therefore π is a finite partial automorphism of M i [s] .Consequently, π = π n,i,i for some n with s = s n,i .We define π ′ := {π n,i,j : j ∈ ω 1 , j ≥ i}.π ′ is a partial isomorphism of A [s] .By our construction, we have M j ⊆ dom π ′ for all j ∈ C n,i , i.e. dom π ⊇ j∈Cn,i M j = ω 1 .The same holds for im π ′ , therefore π ′ is a total automorphism of A [s] .
It remains to show that A has the property 3.1(3).Proof.Let S be a countable subset of A, x, y ∈ S and h : A \ S → ω be a function.Then we define that E(x, y, S, h) is true iff for all m and for all a = (a 1 , . . ., a m ) ∈ A m \ S m the following holds: If (̺ m (a) ∧ max h(a) < m ∧ (∃i, j ∈ {1, . . ., m})(x = a i ∧ y = a j )), then i < j We claim x < y ⇐⇒ (∃S)(∃h)E(x, y, S, h).
Proof of "⇒": Let i ∈ ω 1 be the least ordinal with x, y ∈ M i .Let S := M i .We have M i ❁ A, therefore M i ❁ h A for some h : A \ S → ω.If a ∈ A m \ S m , is weak for h, then ̺ m (a) ⇐⇒ a 1 < a 2 < . . .< a m .Therefore, if x = a i , y = a j and x < y, then i < j.Consequently all tuples in γ i are transformed by g to tuples in γ k .Therefore there exists a function g 0 : κ → κ with g[γ i ] ⊆ γ g0(i) .
g −1 is also an automorphism of ̺, and it is easy to see that the corresponding function g ′ 0 : κ → κ has to be the inverse of g 0 .Consequently, g 0 is a permutation on κ.
But κ, < has only the trivial order automorphism, therefore g 0 = id κ .We obtain g[γ i ] ⊆ γ i and (because g −1 is also an automorphism) g −1 [γ i ] ⊆ γ i , i.e. g[γ i ] = γ i .But then, g is an automorphism for all relations in M , and therefore also for all relations in Q.This yields Aut{̺} ⊆ Aut Q, and this finishes the proof.
As a consequence of this Lemma, every possible automorphism group appears already as the automorphism group of an at most countable set of relations.(m i ∈ ω \ {0}), such that for all (̺ i ) 1 i∈ω ∈ 1 i∈ω Rel (mi) (A) and all g ∈ Sym(A) we have If Q ⊆ Rel(A), then Q ω−inv is the closure of Q under all invariant operations with countable arity, and Q ω−inv , is the least set of relations which is closed under all invariant operations with countable arity and -closed.
(Of course, ω−inv and ω−inv , are closure operators.)Similar as in 2.13 and 2.14, we can verify the following properties:  In particular, a set R ⊆ Rel(A) is Galois closed if and only if it is -closed and closed under all invariant operations with countable arity, R = R ω−inv , .For all Q ⊆ Rel(A) holds Q ω−inv , = sInv Aut Q.

Lemma 2 . 5 .
m and the relation a ∼ R b : ⇐⇒ a ∈ Γ R (b) is an equivalence relation.In this case, R (m) is an atomic Boolean algebra.Note that a ∼ R b iff there is no relation ̺ ∈ R separating a from b. (5) If R 1 and R 2 are -closed and closed under complementation, then R 1 = R 2 if and only if Γ R1 (a) = Γ R2 (a) for all m and all a ∈ A m .(6) Γ sInv G (a) = {g(a) | g ∈ G group }, where G group is the subgroup of Sym(A), generated by G. Proof.(1)-(5) are direct consequences of the Definitions.For (6), we first note that {g(a) | g ∈ G group } contains a and is strongly invariant for all g ∈ G. Therefore Γ sInv G (a) ⊆ {g(a) | g ∈ G group }.On the other hand, sInv G is -closed, therefore a ∈ Γ sInv G (a) ∈ sInv G, and every relation with these properties must contain all g(a) with g ∈ G group .Consequently also Γ sInv G (a) ⊇ {g(a) | g ∈ G group }.Let R ⊆ Rel(A) be -closed and closed under complementation.Then R = sInv Aut R if and only if for all m and all a, b ∈ A m with a ∼ R b there exists an automorphism g ∈ Aut R with b = g(a).
) The full countable bipartite graph: (A ∪ B; ̺), where A and B are disjoint countable sets, and ̺ = (A × B) ∪ (B × A).(3)The countable random graph.(See e.g.[4, 6.4.4].)Each of these structures M = (M ; ̺) has the following properties:(a) T h(M), the first order theory of M , is ω-categorical.(b)All unary first order formulas ϕ(x) are equivalent (mod T h(M )) to x = x or to x = x, i.e., the only subsets of M that are first order definable without parameters are the empty set and the whole model.(c) For any uncountable cardinal κ there is a model M κ of cardinality κ such that the set ̺ * := {x : The set {y : ̺(x, y)} is countable} is neither empty nor the full model.

Definition 3 . 5 .
Let M = (M ; (̺ m ) 1 m∈ω ) be a model, let h : M •→ ω be a partial function, n ∈ ω \ {0} and let a = (a 1 , . . ., a n ) ∈ M n .We say that a is a weak n-tuple for h if dom h ∩ {a 1 , . . ., a n } = ∅ and max h(a) := max{h(a i ) | a i ∈ dom h} < n.All other n-tuples are called strong for h.Now let N = (N ; (σ m ) 1 m∈ω ) and M = (M ; (̺ m ) 1 m∈ω ) be models with N ⊂ M ⊆ ω 1 .Let h : M •→ ω be a partial function with dom h = M \ N .We write N ❁ h M if (1) N M (N is a submodel of M ), and

Lemma 4 . 2 .Definition 4 . 3 .
For every set R ⊆ Rel(A) there exists an at most countable setR 0 ⊆ Rel(A) with Aut R = Aut R 0 .Moreover, if R is a -closed Krasner algebra, then we can choose R 0 ⊆ R. Proof.If R is not closed under C, then we put R ′ := R ∪ {Cσ | σ ∈ R}.Then Aut R = Aut R ′ , therefore we can assume w.l.o.g. that R is closed under complementation.By 4.1 there are relations ̺ m ∈ Rel (2m) (A) with Aut R (m) = Aut{̺ m }.Consequently: Aut R = 1 m∈ω Aut R (m) = 1 m∈ω Aut{̺ m } = Aut{̺ m | 1 m ∈ ω}The second part in 4.1 implies that the ̺ m can be chosen from the -closed Krasner algebra, generated by R. Now we define our additional operations.An invariant operation with countable arity is an operation of the form F :1 i∈ω Rel (mi) (A) → Rel (m) (A)

Lemma 4. 4 . ( 1 )
If R ⊆ Rel(A) is Galois closed, R = sInv Aut R, then R is -closed and closed under all invariant operations with countable arity, R = R ω−inv , .(2) If Q ⊆ Rel(A) is countable or finite, then Q ω−inv = sInv Aut Q.Now we can formulate our characterization of the Galois closed sets of relations.

Theorem 4 . 5 .
Let R be a -closed Krasner algebra.Then R is Galois closed, R = sInv Aut R, if and only if sInv Aut R 0 ⊆ R for every countable subset R 0 of R.
we have the closure operators KA , inv , KA, , inv , and sInv Aut.The operators KA and inv are algebraic, the other operators are not algebraic if A is infinite.For all Q ⊆ Rel(A) we have