We construct a Borel maximal cofinitary group.

A first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega $-categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal {M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal {M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas’ conjecture for the class $\mathcal {M}$.

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

Given two structures ${\cal M}$ and ${\cal N}$ on the same domain, we say that ${\cal N}$ is a reduct of ${\cal M}$ if all $\emptyset$-definable relations of ${\cal N}$ are $\emptyset$-definable in ${\cal M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are ${\aleph _0}$-categorical, determining their reducts is equivalent to determining the closed supergroups G ≤ Sym(ℕ) of their automorphism groups.

A consequence of the classification is that there are ${2^{{\aleph _0}}}$ pairwise noninterdefinable Henson digraphs which have no proper nontrivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are ${2^{{\aleph _0}}}$ pairwise nonconjugate maximal-closed subgroups of Sym(ℕ). By the reconstruction results of Rubin, these groups are also nonisomorphic as abstract groups.

Let ($\rm L$;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of ($\rm L$;C), i.e., the structures with domain $\rm L$ that are first-order definable in ($\rm L$;C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of ($\rm L$;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.

A sufficient criterion is found for certain permutation groups $G$ to have uncountable strong cofinality, that is, they cannot be expressed as the union of a countable, ascending chain $(H_i)_{i\in\o}$ of proper subsets $H_i$ such that $H_iH_i \subseteq H_{i+1}$ and $H_i\,{=}\,H_i^{-1}$. This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$. This basic tool comes from a recent paper by Bergman on generating systems of the infinite symmetric groups, which is discussed in the introduction. The main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. Thus the result also unifies various known results about cofinalities. A notable example is the group BSym ($\Q$) of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality.

It is shown that the countable saturated discrete linear ordering has the small index property, but that the countable 1-transitive linear orders which contain a convex subset isomorphic to ${\Bbb Z}^2$ do not. Similar results are also proved in the coloured case.

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be $\ell$-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups.

THEOREM. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be$\ell$-embedded in a finitely presented lattice-ordered group.

If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$.

COROLLARY. $D(\xi)$can be$\ell$-embedded in a finitely presented lattice-ordered group if and only if$\xi$is a recursive real number.

Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘$\ell$-algebraic’ in that it can be captured by finitely many relations in this language.

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.

Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

Let be a finite set of finite tournaments. We will give a necessary and sufficient condition for the -free homogeneous directed graph to be divisible. That is, that there is a partition of into two classes such that neither of them contains an isomorphic copy of .