
Dugald MacphersonUniversity of Leeds · School of Mathematics
Dugald Macpherson
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Publications (144)
We explore two constructions of oligomorphic Jordan permutation groups preserving a `limit of betweenness relations' and a `limit of $D$-relations', from \cite{bhattmacph2006jordan} and \cite{almazaydeh2021jordan} respectively. Several issues left open in \cite{almazaydeh2021jordan} are resolved. In particular it is shown that the `limit of $D$-rel...
We develop a general framework (multidimensional asymptotic classes, or m.a.c.s) for handling classes of finite first order structures with a strong uniformity condition on cardinalities of definable sets: The condition asserts that definable families given by a formula \phi(x,y) should take on a fixed number n_\phi of approximate sizes in any M in...
VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of dependent...
We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these...
We construct via Fra\"iss\'e amalgamation an $\omega$-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a `limit of $D$-relations'. The construction is based on a semilinear order whose elements are labelled by sets carrying a $D$-relation, with strong coherence conditions gover...
VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of \emph{depe...
We classify countable metrically homogeneous graphs of diameter 3.
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite...
We call a non-identity element of a permutation group irreducible if it cannot be written as a product of non-identity elements of disjoint support. We show that it is indeed possible for a sublattice subgroup of \(\mathrm{Aut}(\mathbb{R},\leq )\) to have no irreducible elements and still be transitive on the set of pairs α < β in \(\mathbb{R}\). T...
We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of {\em all} open subgroups, then the first order theory of such a structure is NIP (that is, does not have the ind...
We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of {\em all} open subgroups, then the first order theory of such a structure is NIP (that is, does not have the ind...
We explore a notion of pseudofinite dimension, introduced by Hrushovski and
Wagner, on an infinite ultraproduct of finite structures. Certain conditions on
pseudofinite dimension are identified that guarantee simplicity or
supersimplicity of the underlying theory, and that a drop in pseudofinite
dimension is equivalent to forking. Under a suitable...
Model theory is a branch of mathematical logic dealing with mathematical structures (models) from the point of view of first order logical definability. Although comparatively young, it is now well established, its major textbooks including [6, 17, 34, 43, 53]. A typical goal of model theory is to build, study and classify mathematical universes in...
LachlanA. H.. On countable stable structures which are homogeneous for a finite relational language. Israel journal of mathematics, vol. 49 (1984), pp. 69–153. CherlinG. and LachlanA. H.. Stable finitely homogeneous structures. Transactions of the American Mathematical Society, vol. 296 (1986), pp. 815–850. - Volume 58 Issue 1 - Dugald Macpherson
Answering a question of Junker and Ziegler, we construct a countable first
order structure which is not omega-categorical, but does not have any proper
non-trivial reducts, in either of two senses (model-theoretic, and
group-theoretic). We also construct a strongly minimal set which is not
omega-categorical but has no proper non-trivial reducts in...
Let G,H be closed permutation groups on an infinite set X, with H a subgroup
of G. It is shown that if G and H are orbit-equivalent, that is, have the same
orbits on the collection of finite subsets of X, and G is primitive but not
2-transitive, then G=H.
We show that any pseudofinite group with NIP theory and with a finite upper
bound on the length of chains of centralisers is soluble-by-finite. In
particular, any NIP rosy pseudofinite group is soluble-by-finite. This
generalises, and shortens the proof of, an earlier result for stable
pseudofinite groups. An example is given of an NIP pseudofinite...
We give an example of an imaginary defined in certain valued fields with
analytic structure which cannot be coded in the `geometric' sorts which suffice
to code all imaginaries in the corresponding algebraic setting.
A famous result by Jeavons, Cohen, and Gyssens shows that every Constraint Satisfaction Problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the so-called universal algebraic approach to a systematic theory of tractability and hardness in finite domain con...
We study the Vapnik-Chervonenkis (VC) density of definable families in
certain stable first-order theories. In particular we obtain uniform bounds on
VC density of definable families in finite U-rank theories without the finite
cover property, and we characterize those abelian groups for which there exist
uniform bounds on the VC density of definab...
We recast the problem of calculating Vapnik-Chervonenkis (VC) density into
one of counting types, and thereby calculate bounds (often optimal) on the VC
density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal
theories.
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 Fraissé limits of amalgamation classes of finite...
Intended for researchers and graduate students in theoretical computer science and mathematical logic, this volume contains accessible surveys by leading researchers from areas of current work in logical aspects of computer science, where both finite and infinite model-theoretic methods play an important role. Notably, the articles in this collecti...
Let M be a countably infinite first order relational structure which is homogeneous in the sense of Frassé. We show, under the assumption that the class of finite substructures of M has the free amalgamation property, along with the assumption that Aut(M) is transitive on M but not equal to Sym(M), that Aut(M) is a simple group. This generalises re...
A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in both directions. Under the assumption tha...
We consider groups G interpretable in a supersimple finite rank theory T such that Teq eliminates ∃∞. It is shown that G has a definable soluble radical. If G has rank 2, then if G is pseudofinite, it is soluble-by-finite, and partial results are obtained under weaker hypotheses, such as ‘functional unimodularity’
of the theory. A classification is...
A graph is connected-homogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably infinite connected-homogeneous graphs. We prove that if Γ is connected countably infinite and connected-homogeneous then Γ is isomorphic to one of: Lachlan and Woodrow's ultr...
This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many w...
We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper
bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that
every ultraproduct of the family is primitive. A key result is that, in the almost simple case with...
A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ε ℕ and every formula φ(x, ȳ), where ȳ = (y1, ⋯ ,ym): (i) There is a positive constant C and a finite set E ⊂ ℝ>0 such that for every M ε C and ā ε Mm, either |φ(M, ā)| ≤ C, or for some μ ε E, ||φ(M, ā)| - μ|M|| ≤ C|M|1/2. (ii) For every μ ε E, there is an L-fo...
This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense, coming from Baire category).
Reconstruction results give...
The main theorem is that if G is a pseudofinite group with stable theory, then G has a definable normal soluble subgroup of finite index.
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 K of certain definable R-submodules of K (for all n 1). The proof involves the development of a theory of independence for unary types, which play the role of 1-types,...
This result rests on the Lang-Weil estimates for the number of F-rational points of an absolutely irreducible variety defined over the finite field F. The proof uses partial quantifier elimination for pseudofinite fields, derivable from the paper of Ax [2] which introduced pseudofinite fields: any formula (炉 x) is a boolean combination of formulas...
A construction is given of an infinite primitive Jordan permutation group which preserves a 'limit' of betweenness relations. There is a previous construction due to Adeleke of a Jordan group of this kind. The main difference is that in this paper the group arises as the automorphism group of an (N 0-categorical) relational structure. It is 2-trans...
We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable...
We explore analogues of o-minimality and weak o-minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ∅ there are differences. Our main result is a structure theory (with infinitely many doubly transitive examples related to J...
The main theorem is that if G is a Polish group with a comeagre conjugacy class, and G acts without inversions on some tree T, then for every g∈G there is a vertex of T fixed by g. In particular, such a group cannot be written non-trivially as a free product with amalgamation. The same conclusion holds if G is the automorphism group of an ω-categor...
The automorphism group of the random graph has a locally finite subgroup with the same orbits as the whole automorphism group on finite sequences of vertices.
The paper concerns sufficiently saturated structures M over a countable language with a unary predicate P. It is shown that if P(M)is stably embedded and there are no Vaughtian pairs with respect to P, then an infinite group is interpretable over M (in an infinitary sense of ‘interpretable’). Also, it is shown that if M is ω-categorical, f:D→P is a...
A partially ordered set (P, ≤) is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of (P, ≤). Assuming the set-theoretic assumption ⋄(ϰ1), it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not (k + 1)-homogeneous....
The paper concerns suciently saturated structures M over a countable language with a unary predicate P . It is shown that if P (M) is stably embedded and there are no Vaughtian pairs with respect to P , then an in nite group is interpretable over M (in an in nitary sense of `interpretable'). Also, it is shown that if M is !-categorical, f : D ! P i...
We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice
and of the countable atomless generalized Boolean lattice. Also, we show that subgroups of these automorphism groups of index
less than lie between the pointwise and the setwise stabilizer of a finite set.
No-categorical o-minimal structures were completely described by Pillay and Steinhorn (Trans. Amer. Math. Sec. 295 (1986) 565-592), and are essentially built up from copies of the rationals as an ordered set by 'cutting and copying'. Here we investigate the possible structures which an No-categorical weakly o-minimal set may carry, and find that th...
A linearly ordered structure is weakly o-minimal if all of its de-finable 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...
In this paper we develop a structure theory for transitive permutation groups definable in o-minimal structures. We fix an
o-minimal structure M, a group G definable in M, and a set Ω and a faithful transitive action of G on Ω definable in M, and talk of the permutation group (G, Ω). Often, we are concerned with definably primitive permutation grou...
The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting for the model theory of expansions of the reals. Section 2 gives some basics (the Monotonicity and Cell Decomposition Theorems) together with a discussion of dimension...
Certain permutation representations of free groups are constructed by finite approximation. The first is a construction of a cofinitary group with special properties, answering a question of Tim Wall published by Cameron. The second yields, via a method of Kepert and Willis, a totally disconnected locally compact group which is compactly generated...
-categorical o-minimal structures were completely described by Pillay and Steinhorn (Trans. Amer. Math. Soc. 295 (1986) 565–592), and are essentially built up from copies of the rationals as an ordered set by ‘cutting and copying’. Here we investigate the possible structures which an -categorical weaklyo-minimal set may carry, and find that there a...
The notion of a strongly determined type over A extending p is introduced, where p .∈ S(A). A strongly determined extension of p over A assigns, for any model M ⊂)- A, a type q ∈ S(M) extending p such that, if realises q, then any elementary partial map M → M which fixes acleq(A) pointwise is elementary over . This gives a crude notion of independe...
By Frank O. Wagner: 309 pp., £27.95 (US$44.95, LMS Members' price £20.95) isbn 0 521 59839 7 (Cambridge University Press, 1997).
In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly b...
We describe the automorphism group of the quotient of the symmetric group on a countably infinite set by the group of finitary permutations. We show that the outer automorphism group is infinite cyclic. In the proof, we examine the structure of centralizers in this quotient. 1 Introduction In this paper, S will denote Sym(\Omega\Gamma2 the full sym...
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.- Applicati...
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 ⁺ el...
This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributions from many leading international figures in this area. Topics described include automorphism groups of algebraically clos...
This paper gives a proof of the conjecture, but it remains open whether we can take c = 3. We cannot take c = 2, since 1-dimensional affine groups AGL(1; F ) (F an infinite field) are soluble, and 2-transitive in their action on the affine line. Neumann has shown that for finite soluble permutation groups, the number of orbits on triples tends to i...
Introduction This paper is in part a sequel to [5] and [4]. We construct a certain uncountable graph, thereby answering a question on automorphism groups of infinite graphs which was raised in [5]. The method of construction is order-theoretic, and uses ideas from [4]. We first build a certain uncountable totally ordered set, then obtain from it a...
Introduction In this article we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Q p is the field of p-adic numbers and Z p is the ring of p-adic integers. One of these theorems, 3.32 in [2], says that each subanalytic subset of Z p is semialgebraic. This is extended here as follows. Theorem A Let S ` Z m+1...
Introduction Let G be a connected linear algebraic group over an algebraically closed field K of characteristic p 0. In this paper we determine all finite-dimensional irreducible rational KG-modules V such that G has only a finite number of orbits on the set of vectors in V . We shall call such a module a finite orbit module for G. When K = C , the...
Contents 0 Outline of the Notes 1 1 Introduction to covers 3 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Related Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
this paper, we prove the following related but more model-theoretic result. Recall that a (first-order) structure N is interpretable in a structure M if there is a positive integer n, a definable subset A of M n (that is, the solution set of a formula, possibly with parameters), and a definable equivalence relation E on A, such that the domain of N...
this paper more clearly). The second lemma provides a su#cient condition guaranteeing that a definable partial order on a linearly ordered structure can be extended definably to a partial order which is total on a given interval
This survey is in the same area as those by Cherlin and Lachlan in this volume. The purpose is to show how results on finite permutation groups contribute to the structure theory for stable finitely homogeneous structures and for smoothly approximated structures. The model theory of smooth approximation has been developed very much further by Cherl...
We have seen in the last chapter (Lemma 14.6) that for every k ∈ ℕ we can find a k-transitive but not (k + 1)-transitive permutation group acting on a count ably infinite set. It is natural therefore to ask whether, for every positive integer k, there is an infinite Jordan group which is k-transitive but not (k + 1)-transitive. In this chapter, we...
In this chapter we will state some applications of the material contained in this book, most specifically of the classification theorem for infinite primitive Jordan groups. We have already seen many applications of the Fraïssé amalgamation technique in Chapters 14 and 15. We will also state some questions which are of current research interest and...
In this chapter we state (mostly without proof) some basic facts from the theory of groups. We have included only those facts from the general theory that we shall have occasion to use later — the treatment is by no means exhaustive. We refer to any standard textbook on Group Theory for more detailed accounts of the subject in general and for the c...
We have already defined the symmetric and alternating groups in Chapter 2 and have also seen some of their basic properties. We will study some more properties of these groups in this chapter.
In this chapter, we introduce the notion of a Jordan set, and study some properties of groups containing Jordan sets. Most of the material covered in this chapter can be found in Adeleke & Neumann (1996a).
In the last chapter we defined linear betweenness relations, circular (or cyclic) orders and separation relations from a linear order and studied their groups of automorphisms. The automorphism group of a linear order has already been studied in detail in Chapter 9. That of a circular order can be understood best in terms of the linear order obtain...
A group is a permutation group if it has a faithful action on a set (cf. Sec. 2.2). And all groups can be considered to be permutation groups, usually in many different ways. Thus a study of these actions is crucial to the understanding of groups. For a detailed analysis of group actions, and for the theory of groups in terms of group actions see N...
We have already seen the concept of an orbit in Chapter 3. We now describe the related notions of suborbits and orbitals of a permutation group G acting on a set Ω.
In this chapter we shall see an application of most of the concepts introduced so far. Wreath product constructions are very important in the study of permutation groups for a variety of reasons, some of which we shall see later in this chapter.