Groups with right-invariant multiorders

A Cayley object for a group G is a structure on which G acts regularly as a group of automorphisms. The main theorem asserts that a necessary and sufficient condition for the free abelian group G of rank m to have the generic n-tuple of linear orders as a Cayley object is that m > n. The background to this theorem is discussed. The proof uses Kronecker’s Theorem on diophantine approximation. 1 Cayley objects and homogeneous structures The regular representation of a group G is the representation of the group acting on itself by right multiplication. A Cayley object for G is a structure on G admitting the regular representation as a group of automorphisms. The name comes from the fact that a Cayley graph for G is precisely a Cayley object which happens to be a graph. A Cayley object must admit a transitive automorphism group. There is some interest in investigating objects with a high degree of symmetry which are Cayley objects for a group, or (in the other direction) groups which have a given highly symmetric object as a Cayley object. This is the topic of [2]; I refer to that paper for further motivation. All objects here will be relational structures, consisting of a set carrying a collection of relations of various arities. A substructure of a relational structure will always be the induced substructure on a subset, consisting of all instances of each relation such that all arguments lie within the subset. A relational structure M is said to be homogeneous if any isomorphism between finite substructures can be extended to an automorphism of M . This will be our “strong symmetry condition”. ∗ Current address: School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland. 188 PETER J. CAMERON The age of a relational structure M is the class of all finite relational structures of the same type which can be embedded into M . Fräıssé [4] gave a necessary and sufficient condition for a class C of finite structures to be the age of a countable homogeneous structure: (a) C is closed under isomorphism; (b) C is closed under taking substructures; (c) C contains only countably many members up to isomorphism; (d) C has the amalgamation property, that is, given A,B1, B2 ∈ C with embeddings fi : A → Bi for i = 1, 2, there exists C ∈ C and embeddings gi : Bi → C for i = 1, 2 such that the composite embeddings g1f1 and g2f2 agree. Moreover, if these conditions hold, there is a unique countable homogeneous structure M with age C (up to isomorphism). Such a class C is called a Fräıssé class, and M is its Fräıssé limit. We say that C has the strong amalgamation property if the amalgamation can be done without identifying points outside A: that is, if b1 ∈ B2 and b2 ∈ B2 satisfy g1(b1) = g2(b2), then there exists a ∈ A such that bi = fi(a) for i = 1, 2. For example, the class of all finite totally ordered sets is a Fräıssé class; its Fräıssé limit is the ordered set Q, the unique countable dense ordered set without endpoints. I generalise this example in the next section. The homogeneous structure M with age C is characterised by the following extension property : If A,B ∈ C with A ⊆ B and |B| = |A| + 1, then every embedding of A into M can be extended to an embedding of B into M .