We examine properties of generic automorphisms of the random poset, with the goal of explicitly characterizing them. We associate to each automorphism an auxiliary first-order structure, consisting of the random poset equipped with an infinite sequence of binary relations which encode the action of the automorphism. We then explicitly characterize generic automorphisms in terms of properties of this structure. Two notable such properties are ultrahomogeneity, and universality for a certain class of finite structures in this language. As this auxiliary structure seems to be new, we also address some model-theoretic questions. In particular, this structure fails to be saturated, and its theory neither is $ω$-categorical nor admits quantifier-elimination, in contrast to many known ultrahomogeneous structures in finite languages. We also examine orbitals -- order-convex hulls of orbits -- and their use in describing automorphisms. In particular, we introduce and use new orders on the space of orbitals.
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give…
We define the notions of a free fusion of structures and a weakly stationary independence relation. We apply these notions to prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the bounded Urysohn space, the random graph, and the random poset.
This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions im…
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.
For a given partially ordered set (poset) and a given family of mappings of the poset into itself, we study the problem of the description of joint fixed points of this family. Well-known Tarski's theorem gives the structure of the set of joint fixed points of isotone automorphisms on a complete lattice. This theorem has several generalizations (see., e.g., Markowsky, Ronse) that weaken demands on the order structure and upgrade in an appropriate manner the assertion on the structure of the set of joint fixed points. However, there is a lack of the statements similar to Kantorovich or Kleene t…
We give a complete description of the size of the conjugacy classes of the automorphism group of the random graph with respect to Christensen's Haar null ideal. It is shown that every non-Haar null class contains a translated copy of a nonempty portion of every compact set and that there are continuum many non-Haar null conjugacy classes. Our methods also yield a new proof of an old result of Truss.
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss's example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of gr…
We study automorphism groups of randomizations of separable structures, with focus on the $\aleph_0$-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the $\aleph_0$-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and cl…
In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different …
Let $(R, \sim )$ be the Rado graph, $Emb (R)$ the monoid of its self-embeddings, $Π(R)=\{ f[R]: f\in Emb (R)\}$ the set of copies of $R$ contained in $R$, and ${\mathcal I}_R$ the ideal of subsets of $R$ which do not contain a copy of $R$. We consider the poset $( Π(R ), \subset )$, the algebra $P (R)/{\mathcal I _R}$, and the inverse of the right Green's pre-order on $Emb (R)$, and show that these pre-orders are forcing equivalent to a two step iteration of the form $P \ast π$, where the poset $P$ is similar to the Sacks perfect set forcing: adds a generic real, has the $\aleph _0$-covering p…
A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the $k$th level of the hierarchy can consist of all structures having at least $k$ elements which are m…
A relational structure ${\mathbb X}$ is said to be reversible iff every bijective endomorphism $f:X\rightarrow X$ is an automorphism. We define a sequence of non-zero cardinals $\langle κ_i :i\in I\rangle$ to be reversible iff each surjection $f :I\rightarrow I$ such that $κ_j =\sum_{i\in f^{-1}[\{ j \}]}κ_i$, for all $j\in I $, is a bijection, and characterize such sequences: either $\langle κ_i :i\in I\rangle$ is a finite-to-one sequence, or $κ_i\in {\mathbb N}$, for all $i\in I$, $K:=\{ m\in {\mathbb N} : κ_i =m $, for infinitely many $i\in I \}$ is a non-empty independent set, and $\gcd (K…
In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particula…
We investigate possible cardinalities of maximal antichains in the poset of copies $\langle \mathbb P(\mathbb X),\subset \rangle$ of a countable ultrahomogeneous relational structure $\mathbb X$. It turns out that if the age of $\mathbb X$ has the strong amalgamation property, then, defining a copy of $\mathbb X$ to be large iff it has infinite intersection with each orbit of $\mathbb X$, the structure $\mathbb X$ can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum…
A poset ${\mathbb{P}}$ is called reversible iff every bijective homomorphism $f:{\mathbb{P}} \rightarrow {\mathbb{P}}$ is an automorphism. Let ${\mathcal{W}}$ and ${\mathcal{W}} ^*$ denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the form ${\mathbb{P}} =\bigcup _{i\in I}{\mathbb{L}} _i$, where ${\mathbb{L}} _i, i\in I$, are pairwise disjoint linear orders from ${\mathcal{W}} \cup {\mathcal{W}} ^*$. First, if ${\mathbb{L}} _i \in {\mathcal{W}}$, for all $i\in I$, and ${\mathbb{L}} _i \cong α_i =γ_i+n_i\in Ord$, where $γ_…
In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density $κ$, for uncountable $κ$ such that $κ^{<κ}=κ$, is not a universal topological group of weight $κ$. We investigate automorphism groups of other uncountable ultrahomogeneous structures and prove that they are rarely universal topological groups for the corresponding classes. Our list of uncountable ultrahomogeneous structures includes random uncountable graph, tournament, linear order, partial order, group. That is in contrast with similar results obtained for automorphism groups of c…
We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or contains an operation that maps the random graph injectively to an induced subgraph which is a clique or an independent set. As a corollary, our techniques yield a new proof of Simon Thomas' classification of the five closed supergroups of the automorphism group of the random graph; our proof uses different Ramsey-theoretic tools than the one given by Thomas…
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.
A relational structure is called reversible iff every bijective endomorphism of that structure is an automorphism. We give several equivalents of that property in the class of disconnected binary structures and some its subclasses. For example, roughly speaking and denoting the set of integers by ${\mathbb Z}$, a structure having reversible components is reversible iff its components can not be "merged" by condensations (bijective homomorphisms) and each ${\mathbb Z}$-sequence of condensations between different components must be, in fact, a sequence of isomorphisms. We also give equivalents o…
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which…
