1 Introduction

Permutations on the random permutation

Abstract.

The random permutation is the Fraïssé limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure. Equivalently, we classify all structures which have a first-order definition in the random permutation.

The research of Michael Pinsker has been funded through project I836-N23 of the Austrian Science Fund (FWF)

1. Introduction

1.1. Homogeneous permutations and the random permutation.

In a paper in 2002, Peter Cameron regarded finite permutations as two linear orders on a finite set, thereby taking a more “passive” perspective on permutations than the one which views them as bijections [Cam02]. He showed that there exist precisely four Fraïssé classes (in the sense of [Hod97]) of finite permutations in this sense, one of which is the class of all finite structures with two linear orders. The Fraïssé limit of the latter class, which is called the random permutation and which we denote by , therefore is the (up to isomorphism) unique countable homogeneous structure with two linear orders which contains all finite permutations as induced substructures. Both linear orders of the random permutation are isomorphic to the order of the rational numbers, and the random permutation is the result that appears with probability one in the natural random process that constructs both orders independently. From this it becomes clear that the random permutation cannot correspond to a single bijection on its domain : indeed, it represents a double coset in the full symmetric group on , where is any isomorphism from to , and denotes the automorphism group of , for .

1.2. Symmetries of the random permutation.

The random permutation possesses two kinds of obvious symmetries. Firstly, it inherits symmetries of the order of the rational numbers: for example, the structure is obviously isomorphic to , and it is easy to see that likewise is isomorphic to . The symmetries of the order of the rational numbers have been classified by Cameron in a famous paper in 1976 [Cam76]; they are basically composed of two non-trivial symmetries, one of which is reversing the order, and the other one is turning the order cyclically. The second obvious symmetry of is the fact that not only the orders and are isomorphic, but also is isomorphic to .

The symmetries in the above sense of a structure correspond to those subgroups of the full symmetric group of its domain which contain the automorphism group of the structure and which are closed in the topology of pointwise convergence. Combining the two kinds of obvious symmetries of mentioned above, Cameron counted 37 closed supergroups of , and asked whether there were any others, stating the following problem:

Problem 1.1 (Problem 2 in [Cam02], rephrased).

Determine the closed subgroups of which contain .

In this paper, we solve this problem, showing that there exist precisely 39 closed supergroups of . While there turn out be a few groups which had not been considered in [Cam02], some of those counted in that paper actually coincide.

1.3. Reducts and Thomas’ conjecture.

For structures on the same domain, we call a reduct of iff all of its relations and functions have first-order definitions in without parameters. It follows from the theorem of Ryll-Nardzewski, Engeler, and Svenonius (see e.g. [Hod97] for all standard model-theoretic notions and theorems) that if we consider two reducts equivalent iff they are reducts of one another, then the reducts of an -categorical structure correspond precisely to the closed supergroups of the automorphism group . In this correspondence, every reduct of is sent to , defining a subjective map onto the closed supergroups of whose kernel is the above-mentioned equivalence. Since the closed supergroups of form a complete lattice, so do the reducts of up to equivalence, the order being provided by first-order definability.

In 1991, Simon Thomas conjectured that every countable structure which is homogeneous in a finite relational language has only finitely many reducts up to equivalence [Tho91]. At the time, the reducts of only two interesting structures which fall into the scope of the conjecture had been classified: those of the order of the rational numbers (5 reducts) [Cam76] and those of the random graph (5 reducts) [Tho91]. Since then the reducts of the random hypergraphs [Tho96], the random tournament [Ben97], the order of the rationals with a constant [JZ08], and more recently those of the random partial order [PPP11], the -free graphs with a constant [Pon11] and the random ordered graph [BPP13] have been determined, in all cases confirming Thomas’ conjecture. Our classification verifies the conjecture for the random permutation.

1.4. Superpositions of homogeneous structures.

Let be Fraïssé classes of finite structures in disjoint signatures and , respectively, and assume moreover that both classes have strong amalgamation. Then the class of finite structures with signature whose restriction to the signature is an element of for is a Fraïssé class as well. Moreover, the restriction of its Fraïssé limit to the signature is the Fraïssé limit of for . In this situation, we say that is the free superposition of the Fraïssé limits of and . Using this terminology, the random permutation is the free superposition of two copies of the order of the rational numbers.

It was only very recently that the reducts of a freely superposed structure, namely the superposition of the random graph and the order of the rational numbers called the random ordered graph, were classified up to equivalence [BPP13]. Our result is the second such classification. One notable contrast between the situation in [BPP13] and our situation is that the two relations of the random ordered graph are very different, the graph relation being a quite “free” binary relation as opposed to the order relation, which gives rise to some asymmetry; in particular, the two relations cannot be flipped.

In the case of the random permutation, another kind of rather surprising asymmetry appears with respect to possible combinations of the reducts of the two orders. As implied above, one closed supergroup of is the one consisting of all order preserving and all order reversing permutations; another one is the one consisting of all permutations which turn the order cyclically. While the first group can be combined with the corresponding group above to the group consisting of all permutations which either reverse or preserve both orders simultaneously, the groups of cyclic turns have no similar “simultaneous” action – see the discussion in Section 5 for more details.

1.5. Canonical functions and Ramsey theory.

We prove our result using a method originally invented in the context of constraint satisfaction [BP11b, BP] and further developed in [BP11a, BPT13]. Based on so-called canonical functions, this method turned out to be very effective in reduct classifications of homogeneous structures with a Ramsey expansion. First applied to this kind of problem in 2011 to determine the reducts of the random partial order [PPP11], it has since served to find the reducts of the -free graphs with a constant [Pon11] and the random ordered graph [BPP13]. As in the case of the latter structure, we take the approach of first identifying the join irreducible elements of the lattice of closed supergroups of with the help of canonical functions. We then use canonical functions again to prove that every closed supergroup of is a join of these groups, exploiting the fact that is itself a Ramsey structure (cf. Section 3).

1.6. A model of the random permutation.

It is helpful to visualize by means of the following concrete representation of this structure. Let be the rational numbers with the usual order . Call a subset of independent iff for all we have and . Then the following is easily verified using the fact that is, up to isomorphism, uniquely determined by the expansion property [Hod97].

Fact 1.2.

Let be any dense and independent subset of . Then setting iff for , we have that is a model of (the theory of) .

1.7. Acknowledgements.

The first author would like to thank her advisor Ágnes Szendrei for her continued guidance and support, and for introducing her to the second author and this problem. The second author is indebted to Igor Dolinka and Dragan Mašulović for drawing his attention to Peter Cameron’s question, as well as for valuable discussion and generous hospitality during his visit at the University of Novi Sad. He would also like to thank Ágnes Szendrei and Keith Kearnes for their equally generous hospitality during his visit at the University of Colorado at Boulder.

2. The Reducts of

2.1. Generators of closed supergroups of .

With the aim of listing the closed supergroups of , we shall now provide a finite set of permutations on such that every closed supergroup of is generated by a subset of that set, in the following sense.

Definition 2.1.

Let be a set of permutations on , and let be a closed permutation group on . We say that generates (over ) iff is the smallest closed permutation group that contains ; in that case, we write . We always assume to be present in the generating process, and will not mention it explicitly. When , then we also write for .

The elements of are precisely those permutations of with the property that for all finite there exists a term function over the set which agrees with on . Here, terms are composites of elements of and of inverses of such elements.

As noted before, the structures , , and are all isomorphic to . Let , and be isomorphisms from to these structures: that is, reverses while preserving , does the same with the roles of the two orders interchanged, and reverses both orders. Moreover, is isomorphic to ; let be an isomorphism.

In the model of provided in Fact 1.2, we can visualize these permutations as follows. Observe that if is dense and independent, then there exist automorphisms of such that maps bijectively onto . Moreover, if automorphisms of are so that maps bijectively onto , then there exists such that . Hence, every function with the property that it sends bijectively onto a set which is dense and independent induces permutations on of the form , and any two permutations of this form are equivalent for our purposes since they generate the same closed groups.

In this construction, is induced by the mapping from to which sends any to ; we may thus say that geometrically, corresponds to the mapping on . Similarly, corresponds to , and to , which is just the composite of the preceding two functions. The function is geometrically nothing else but .

We use our model of in order to define more permutations. Let be an irrational number, and let be any function which sends the interval bijectively onto whilst preserving the order on and . Then is a permutation of which induces a permutation on as described above – we denote this permutation by . It is straightforward to see that the closed group generated by such a function is independent of , and we will thus write whenever there is no need to refer to explicitly. Similarly, we define functions and .

2.2. Closed supergroups of .

Recall that is isomorphic to the order of the rational numbers, and that the closed supergroups of the automorphism group of that order have been classified [Cam76]. In our context, that classification can be stated as follows.

Theorem 2.2 (Cameron [Cam76]).

The closed supergroups of are precisely the following:

  1. ;

  2. ;

  3. ;

  4. ;

  5. .

Of course, the theorem for is similar. If we wish to see these groups as automorphism groups of reducts of , then the following relations on are suitable. For , set

  • ;

  • ;

  • .

Figure 1. Closed supergroups of .
Corollary 2.3 (Cameron [Cam76]).

The closed supergroups of are precisely the following:

  1. ;

  2. ;

  3. ;

  4. .

  5. .

The groups in Theorem 2.2 and Corollary 2.3 are listed in the same order.

2.3. Join irreducible closed supergroups of .

Arbitrary intersections of closed permutations groups on yield closed permutation groups. Therefore, the closed permutation groups on form a complete lattice with respect to inclusion, and the closed supergroups of form an interval therein. We now provide the set of all completely join irreducible elements of the lattice , i.e., of all elements of which are not the (in theory, possibly infinite) join of other groups in .

Definition 2.4.

Let consist of the following groups:

  1. ;

  2. ;

  3. ;

  4. ;

  5. ;

  6. ;

  7. ;

  8. ;

  9. ;

  10. .

We are going to prove the following theorem, which implies that the closed permutation groups which properly contain are precisely the joins of groups in . As a consequence, it follows that there are at most closed supergroups of .

Theorem 2.5.

Let be a closed group and let be such that . Then there exists a group such that and .

Corollary 2.6.

Let be a closed group. Then is the join of elements of . In particular, is finite.

By systematically investigating the joins of elements of , we then obtain that there exist precisely 39 distinct closed supergroups of , and determine the exact shape of . In order to show a compact picture of , we name the elements of as follows. First those which we know from the classification of the symmetries of the order of the rational numbers…

Letter a b c d e
Group

…and then those which we get by switching the orders, or by completely ignoring one of the orders. In Figure 2, each group in is labeled by a minimal set of elements of whose join it equals.

f g h i j
Theorem 2.7.

The lattice of closed supergroups of has 39 elements, and the shape as represented in Figure 2.

e

f

a

b

g

c

d

ac

be

ef

ad

ab

bd

de

h

bc

cd

bde

abc

abd

acd

bcd

bf

bg

af

i

j

abcd

bef

bh

di

bj

aj

ci

abf

abj

cdi

Figure 2. The lattice of closed supergroups of .

2.4. Organization of the paper.

In the following section (Section 3), we provide the Ramsey-theoretic preliminaries for the proof of Theorem 2.5, consisting of a Ramsey-type statement for the class of finite permutations, and the method of canonical functions. After that, in Section 4, we give the proof of Theorem 2.5. In the final section (Section 5), we show which joins of element in actually coincide, from which we obtain the precise size and shape of , proving Theorem 2.7. It is also there that we discuss the shape of in more detail, and compare it with Peter Cameron’s count from [Cam02].

3. Ramsey-theoretic Preliminaries.

3.1. Ramsey structures.

Our proof exploits a Ramsey-type property of , as in the following definition.

Definition 3.1.

Let be a countable relational structure with domain . For any structure in the language of , write for the set of all induced substructures of which are isomorphic to . Then is called a Ramsey structure iff for all finite induced substructures of , all induced substructures of , and all there exists such that the restriction of to is constant.

For example, the order of the rational numbers is Ramsey: this fact is easily seen to be equivalent to Ramsey’s theorem. It follows that is Ramsey, since it is the free superposition of two copies of a homogeneous relational Ramsey structure whose finite substructures are rigid and have strong amalgamation. That such superpositions are Ramsey has been proven in [Bod] using infinitary methods, and later in [Sok] using finite combinatorics. The fact that is Ramsey was, however, proven before in [BF13] and independently in [Sok10].

Fact 3.2.

is a Ramsey structure.

3.2. Canonical functions.

The fact that is a relational homogeneous Ramsey structure implies that distinct closed supergroups of can be distinguished by so-called canonical functions. This has been observed in [BP11a, BPT13], and will be our method for proving our main result.

Definition 3.3.

Let be a structure, and let be an -tuple of elements in . The type of in is the set of first-order formulas with free variables that hold for in .

Definition 3.4.

Let and be structures. A type condition between and is a pair , such that is the type on an -tuple in and is the type of an -tuple in , for some . A function satisfies a type condition iff the type of in equals for all -tuples in of type .

A behavior is a set of type conditions between and , and a function has behavior iff it satisfies all type conditions in .

Definition 3.5.

Let and be structures. A function is canonical iff for every type of an -tuple in there is a type of an -tuple in such that satisfies the type condition . That is, canonical functions send -tuples of the same type to -tuples of the same type, for all .

Note that any canonical function induces a function from the types over to the types over . The canonical functions that are needed are in general not permutations: they fail to be surjective. In fact, we obtain our classification of the closed supergroups of by an analysis of the closed transformation monoids of injective functions on which contain . Here, just as for permutation groups, “closed” means closed in the topology of pointwise convergence, i.e., the topology of the product space , where is taken to be discrete. The fact that we have to leave the realm of permutation groups necessitates the following definition.

Definition 3.6.

Let . We say that mon-generates a function (over ) iff is contained in the smallest closed submonoid of which contains . In other words, this is the case iff for every finite subset there exist and such that agrees with on . In this paper, when we write that mon-generates a function then we always mean “over ”. For functions we shall say that mon-generates rather than mon-generates .

Our proof relies on the following proposition which is a consequence of [BP11a, BPT13] and the fact that is a Ramsey structure. For , let denote the structure obtained from by adding the constants to the language.

Proposition 3.7.

Let be any injective function, and let . Then mon-generates an injective function such that

  • agrees with on ;

  • is canonical as a function from to .

Note that any two canonical functions from to with equal behavior, i.e., which satisfy the same type conditions, mon-generate one another: this is a consequence of the homogeneity of . Thus, for a fixed choice of , there are essentially only finitely many distinct canonical functions, and they are essentially finite objects since they are determined by their behavior.

Our proof of Theorem 2.5 uses the following idea: if and , then there exist such that no function in agrees with on . Let be the canonical function mon-generated by by virtue of Proposition 3.7; then is not mon-generated by . By analyzing the possible behaviors of , we deduce that mon-generates all functions of some which is not contained in . But then contains , and hence a new element of .

4. The Proof

4.1. All canonical functions from to .

We start by investigating all behaviors of canonical injections from to . As a matter of fact, this will make us rediscover many of the functions presented in Section 2.

Definition 4.1.

Let be canonical. We say that behaves like iff for all we have that the type of equals the type of in .

Definition 4.2.

Define the following binary relations on :

  • ;

  • ;

  • ;

  • .

We call a subset diagonal iff either for all distinct , or for all distinct .

Proposition 4.3.

Let be a closed supergroup of and let be a canonical function mon-generated by . Then either the image of is diagonal and for some , or behaves like one of the following functions:

  1. the identity function on ;

  2. ;

  3. ;

  4. ;

  5. ;

  6. ;

  7. ;

  8. .

Proof.

There are precisely four types of pairs of distinct elements in . Let and be the types of a pair with , , , and , respectively. The behavior of is fully specified by how it behaves on pairs of types and . This gives 16 possible canonical behaviors.

Take any such that the types of and in are and , respectively. Suppose first that and both have type . Then sends to a diagonal set while preserving . Take any with and for all . In order to show that , it suffices to show that there is a function mon-generated by such that for all .

Since is canonical, for all , the pairs and have type . Therefore, the tuples and have the same type in . Since is -categorical, there exists such that for all . Moreover, since is mon-generated by , there exists which agrees with on . Let . Then is a function mon-generated by such that for all , giving that .

If and both have type , then also sends to a diagonal set while preserving . Similarly, if and both have type or , then sends to a diagonal set while reversing . By the same argument as above, holds in these cases as well.

Now suppose has type and has type . Then sends to a diagonal set while preserving . This is also the case if has type and has type . Similarly, if has type and has type , or if has type and has type , then sends to a diagonal set while reversing . Using a similar argument as above, in each of these cases we see that .

This leaves 8 remaining canonical behaviors, corresponding to the behaviors of the 8 functions listed above. ∎

4.2. Canonical behaviors with constants.

Definition 4.4.

Let be distinct. For , define the -th column of , denoted , to be the set of all such that exactly of are less than with respect to the order . Similarly, define the -th row of , denoted , to be the set of all such that exactly of are less than with respect to the order .

For distinct , the infinite orbits of are precisely the sets for all . From our model of in Fact 1.2, the following becomes evident.

Fact 4.5.

Each infinite orbit of is isomorphic to .

By Proposition 3.7, we know that closed groups containing can be distinguished by functions which are canonical from to , for some . Every such function behaves like a canonical function from to on each infinite orbit of , in the following sense.

Definition 4.6.

Let be canonical, let be disjoint, and let be a function. We say that behaves like on iff for all we have that the type of equals the type of in . We say that behaves like between iff for all , we have that the type of equals the type of in .

In this subsection we will show that in a sense, there are only few relevant canonical functions from to . More precisely, we will prove Lemma 4.14 which states that if is a canonical injection from to which behaves like on some infinite orbit of , then either is mon-generated by , or else any closed group which mon-generates contains for some . From this the proof of Theorem 2.5 quickly follows.

Definition 4.7.

Let be a relation, and let . We say that holds iff holds for all .

Definition 4.8.

Let be disjoint. We say that are in diagonal position iff either or holds. For a function , we say that diagonalizes iff are in diagonal position.

Definition 4.9.

Let be subsets of , , and let be a function. We say that

  1. preserves on iff implies , for all ;

  2. reverses on iff implies , for all ;

  3. preserves between and iff implies , for all ;

  4. reverses between and iff implies , for all .

Lemma 4.10.

Let and let be canonical. Let with , and let be infinite orbits of with . If behaves like on or on , then

  1. either preserves or reverses between and ;

  2. either preserves on or diagonalizes and .

Proof.

Without loss of generality, suppose and that behaves like on . If and are in diagonal position, then all pairs with and have the same type in . Therefore, all pairs with and have the same type in . It follows that and are in diagonal position, verifying (2), and moreover (1) holds. We may thus henceforth assume that and are not in diagonal position.

We first show (1). If does not preserve between and , then there exist and such that ; without loss of generality we may assume . Pick such that and . Since preserves on , . So, by transitivity, . Since is canonical it follows that reverses between and . Hence, either preserves or reverses between and .

To see (2), suppose first that violates between and . Without loss of generality, there exist with and . Take any . If , then since is canonical, . If , pick with . Then by the above, . Moreover, since preserves on , . So, by transitivity, . Hence, . This together with (1) shows that and are in diagonal position.

Now assume that preserves between and . Take any , such that . Since preserves between and , . Hence, preserves on . ∎

Lemma 4.11.

Let be a closed supergroup of . Let and let be a function mon-generated by which is canonical as a function from to . Suppose diagonalizes infinite orbits of which are not in diagonal position. Then , for some .

Proof.

Without loss of generality, suppose and . We claim that mon-generates a function which behaves like on and , and such that holds.

Since is canonical as a function , it behaves like a canonical function from to on each of and . Say behaves like on and on . Then (and similarly ) is mon-generated by : any self-embedding of whose range is contained in is mon-generated by , and behaves like . Therefore, by Proposition 4.3, either for some , or else behaves like one of or on . We may thus assume the latter holds, for both and . Note that each of and behaves like on .

Let be such that . Then there is a self-embedding of which sends and into and , respectively. To see this, let be a -downward closed subset of without a -largest element which contains and is disjoint from . Let denote the structure obtained from by adding the unary relation to the language. Let be the structure induced by , and let denote the structure obtained from by adding the unary relation to the language. Then the structures and are isomorphic, and any isomorphism from the first to the latter is an embedding of as desired.

If , then by the above there exists an embedding of such that and . Let . Then since and behave like on , is a function mon-generated by which behaves like on and , and such that holds.

If on the other hand , then pick an embedding of such that and . Let . Since and both behave like on , is a function mon-generated by which behaves like on and , and such that holds, thus proving our claim.

Take any with for all . By homogeneity of , for each there exists such that for all and for all . Then diagonalizes the sets and while preserving on . By repeated applications of such functions, we obtain a function mon-generated by such that holds for all . It then follows by homogeneity of and topological closure that mon-generates a canonical function from to whose image is a diagonal set, and Proposition 4.3 implies that contains for some . ∎

Lemma 4.12.

Let be a closed supergroup of . Let be distinct and let be a function mon-generated by which is canonical as a function from to . Suppose behaves like on some infinite orbit of . Then behaves like on all infinite orbits of , or else , for some .

Proof.

Let be an infinite orbit of on which behaves like . Let be an orbit in the same column as ; without loss of generality, suppose . Then by Lemmas 4.10 and 4.11, either , for some , or preserves on . We may thus assume the latter holds. Therefore, either behaves like or on , and like or between and .

We claim that if behaves like on , then contains for some . To see this, observe first that in that situation, mon-generates