Dugald Macpherson

A survey of homogeneous structures

A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as Fraisse limits of amalgamation classes of finite structures. The subject has connections to model theory, to permutation group theory, to combinatorics (for example through combinatorial enumeration, and through Ramsey theory), and to descriptive set theory. Recently there has been a focus on connections to topological dynamics, and to constraint satisfaction. The article discusses connections between these topics, with an emphasis on examples, and on special properties of an amalgamation class which yield important consequences for the automorphism group.

Weakly o-minimal structures and real closed fields

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

Definable sets in algebraically closed valued fields: elimination of imaginaries

Abstract It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.

On variants of o-minimality

The hypotheses of o-minimality and strong minimality have yielded rich modeltheoretic consequences. The models in each of these classes are characterized by the property that the definable sets in one variable in any elementarily equivalent structure are exactly those which are quantifier-free definable in some “basic” reduct of the given structure. In the case of o-minimal@ the reduct is a total order, and in strong minimal&y it is just equality. Our goal here is to begin to investigate what results hold if all the structures to be considered are appropriately “minimal” expansions of some other basic relational structure. We are guided in our work by the criteria that the basic structures have mathematically interesting minimal expansions and that the class of minimal expansions has some reasonable model theory analogous to that available in the motivating contexts. Our choice of basic structures is influenced by some as yet unpublished work on Jordan groups, due to Adeleke, Neumann, and the first author [l-3,20]. We now sketch this work in some detail for readers unfamiliar with it. Let (G;X) be a permutation group, that is, G is a permutation group on a set X. If A cX, then GcA), respectively GIAl, denotes the pointwise, respectively setwise, stabilizer of A in G. The permutation group is called primitive if there does not exist a non-trivial G-invariant partition of X. It is said to be k-transitive, where k E N, if G is transitive on the set of ordered k-tuples of distinct elements of X, and is called highly transitive if it is k-transitive for all k E N. A subset A c X is a Jordan set for (G;X) if IAl > 1 and GCXiA) is transitive on A. The set A is called a proper Jordan set

A version of o -minimality for the p -adics

In this paper we formulate a notion similar to o -minimality but appropriate for the p -adics. The paper is in a sense a sequel to [11] and [5]. In [11] a notion of minimality was formulated, as follows. Suppose that L, L + are first-order languages and + is an L + -structure whose reduct to L is . Then + is said to be -minimal if, for every N + elementarily equivalent to + , every parameterdefinable subset of its domain N + is definable with parameters by a quantifier-free L -formula. Observe that if L has a single binary relation which in is interpreted by a total order on M , then we have just the notion of strong o-minimality , from [13]; and by a theorem from [6], strong o -minimality is equivalent to o -minimality. If L has no relations, functions, or constants (other than equality) then the notion is just strong minimality . In [11], -minimality is investigated for a number of structures . In particular, the C-relation of [1] was considered, in place of the total order in the definition of strong o -minimality. The C -relation is essentially the ternary relation which naturally holds on the maximal chains of a sufficiently nice tree; see [1], [11] or [5] for more detail, and for axioms. Much of the motivation came from the observation that a C -relation on a field F which is preserved by the affine group AGL(1, F ) (consisting of permutations ( a,b ) : x ↦ ax + b , where a ∈ F \ {0} and b ∈ F ) is the same as a non-trivial valuation: to get a C -relation from a valuation ν, put C ( x;y,z ) if and only if ν( y − x ) y − z ).

Cell decompositions of C-minimal structures

Abstract C-minimality is a variant of o-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for C-minimal structures is proved, and a notion of dimension is introduced. It is shown that C-minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has the exchange property, and for definable sets dimension coincides with the rank obtained from algebraic closure.

Model theory with applications to algebra and analysis

Preface List of contributors 1. Conjugacy in groups of finite Morley rank Olivier Frecon and Eric Jaligot 2. Permutation groups of finite Morley rank Alexandre Borovik and Gregory Cherlin 3. A survey of asymptotic classes and measurable structures Richard Elwes and Dugald Macpherson 4. Counting and dimensions Ehud Hrushovski and Frank Wagner 5. A survey on groups definable in o-minimal structures Margarita Otero 6. Decision problems in algebra and analogues of Hilberts tenth problem Thanases Pheidas and Karim Zahidi 7. Hilberts tenth problem for function fields of characteristic zero Kirsten Eisentrager 8. First-order characterization of function field invariants over large fields Bjorn Poonen and Florian Pop 9. Nonnegative solvability of linear equations in ordered Abelian groups Philip Scowcroft 10. Model theory for metric structures IItai Ben Yaacov, Alexander Berenstein, C. Ward Henson and Alexander Usvyatsov.

Notes on infinite permutation groups

Some group theory.- Groups acting on sets.- Transitivity.- Primitivity.- Suborbits and orbitals.- More about symmetric groups.- Linear groups.- Wreath products.- Rational numbers.- Jordan groups.- Examples of Jordan groups.- Relations related to betweenness.- Classification theorems.- Homogeneous structures.- The Hrushovski construction.- Applications and open questions.

ONE-DIMENSIONAL ASYMPTOTIC CLASSES OF FINITE STRUCTURES

A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula φ(x, y), where y = (y 1 ,...,y m ): (i) There is a positive constant C and a finite set E C R >0 such that for every M ∈ C and a ∈ M m , either |φ(M, a)| < C, or for some μ, ∈ E, ||φ(M, a)|-μ|M|| ≤ C|M| 1 2 . (ii) For every μ ∈ E, there is an L-formula φ μ (y), such that φ μ (M m ) is precisely the set of a ∈ M m with ||φ(M, a)| - μ|M|| ≤ C|M| 1 2 . One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.

Omega-categoricity, relative categoricity and coordinatisation

L and L − are countable first-order languages, L − ⊆L, and one of the symbols which is in L but not in L − is a 1-ary relation symbol P. The theory T is a complete theory in L with infinite models. We assume that if B is any model of T, the set P B of elements of B which satisfy P(x) is the domain of a substructure of the reduct B|L − of B to L − ; we write P(B) for this substructure. The paper will discuss two main questions about relative categoricity. Suppose T is relatively categorical and B is a model of T. Then we ask when the following hold: (1) B is explicitly definable in terms of P(B). (2) Every element of B is algebraic over P(B)