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MULTIDIMENSIONAL EXACT CLASSES, SMOOTH APPROXIMATION AND BOUNDED 4-TYPES
Abstract
In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class ( R -mec), a special kind of multidimensional asymptotic class ( R -mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language $\mathcal {L}$ and any positive integer d the class $\mathcal {C}(\mathcal {L},d)$ of all finite $\mathcal {L}$ -structures with at most d 4-types is a polynomial exact class in $\mathcal {L}$ , where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.
... As L is fixed, set-homogeneity ensures that there is d ∈ N such that for each M ∈ C, Aut(M ) has at most d orbits on M 4 . The result now follows from Theorem 4. 4.1 of [39], which is based on results from [14]. It is almost immediate from the definition of smooth approximation that if a family F i smoothly approximates an infinite structure M i then M i will itself be set-homogeneous. ...
A $k$-uniform hypergraph $M$ is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs $U,V$ are isomorphic there is $g\in Aut(M)$ with $U^g=V$; the hypergraph $M$ is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous $k$-uniform hypergraphs which are not homogeneous (two with $k=3$, one with $k=4$, and one with $k=6$). Evidence is also given that these may be the only ones, up to complementation. For example, for $k=3$ there is just one countably infinite $k$-uniform hypergraph whose automorphism group is not 2-transitive, and there is none for $k=4$. We also give an example of a finite set-homogeneous 3-uniform hypergraph which is not homogeneous.
We study ultraproducts of finite residue rings \(\prod \nolimits _{n\in {\mathbb {N}}} {\mathbb {Z}}/n{\mathbb {Z}}\diagup {\mathcal {U}} \) where \({\mathcal {U}}\) is a non-principal ultrafilter. We find sufficient conditions of the ultrafilter \({\mathcal {U}}\) to determine if the resulting ultraproduct \(\prod \nolimits _{n\in {\mathbb {N}}} {\mathbb {Z}}/n{\mathbb {Z}}\diagup {\mathcal {U}}\) has simple, NIP, \(\mathrm {NTP}_{2}\) but not simple nor NIP, or \(\mathrm {TP}_{2}\) theory, noting that all these four cases occur.
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 assumption, a
measure-theoretic condition is shown to be equivalent to local stability. Many
examples are explored, including vector spaces over finite fields viewed as
2-sorted finite structures, and homocyclic groups. Connections are made to
products of sets in finite groups, in particular to word maps, and a
generalization of Tao's algebraic regularity lemma is noted.
Let L L be a finite relational language and Hom ( L , ω ) \operatorname {Hom}(L,\omega ) denote the class of countable L L -structures which are stable and homogeneous. The main result of the paper is that there exists a natural number c ( L ) c(L) such that for any transitive M ∈ Hom ( L ; ω ) \mathcal {M} \in \operatorname {Hom}(L;\omega ) , if E E is a maximal 0 0 -definable equivalence relation on M \mathcal {M} , then either | M / E | > c ( L ) |\mathcal {M}/E| > c(L) , or M / E \mathcal {M}/E is coordinatizable. In an earlier paper the second author analyzed certain subclasses Hom ( L , r ) ( r > ω ) \operatorname {Hom}(L, r)\ (r > \omega ) of Hom ( L , ω ) \operatorname {Hom}(L,\omega ) for all sufficiently small r r . Thus the earlier analysis now applies to Hom ( L , ω ) \operatorname {Hom}(L,\omega ) .
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-formula φμ(ȳ), such that φμ(Mm) is precisely the set of ā ε Mm with ||φ(M, ā)| - 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.
A tournament $T$ is called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of $T$. It is shown that there are precisely three homogeneous tournaments of power $\aleph_0$. Some analogous results for 2-tournaments are obtained.
We show that definable sets of finite S
1-rank in algebraically closed fields with an automorphism can be measured.
We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.
Introduction and Background The note contains the details of an assertion made in [1] to the eect that elds admitting a nontrivial strong ordered Euler characteristic are quasinite. In this section we recall the relevant denitions and in the next section we complete the proof. Recall that a eld K is quasinite if K is perfect and its absolute Galois group is isomorphic to the pronite completion of Z. In particular, a nite eld is quasinite. A strong ordered Euler characteristic on the eld K is a function : Def(K) ! R from the set of denable (in the language of rings) subsets of (any Cartesian power) of K to a partially ordered ring R having image amongst the nonnegative elements of R and satisfying (X) = (Y ) for X and<
This book is an up-to-date introduction to simple theories and hyperimaginaries, with special attention to Lascar strong types and elimination of hyperimaginary problems. Assuming only knowledge of general model theory, the foundations of forking, stability and simplicity are presented in full detail. The treatment of the topics is as general as possible, working with stable formulas and types and assuming stability or simplicity of the theory only when necessary. The author offers an introduction to independence relations as well as a full account of canonical bases of types in stable and simple theories. In the last chapters the notions of internality and analyzability are discussed and used to provide a self-contained proof of elimination of hyperimaginaries in supersimple theories.
This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.
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 collection emphasize points of contact and connections between finite and infinite model theory in computer science that may suggest new directions for interaction. Among the topics discussed are: algorithmic model theory, descriptive complexity theory, finite model theory, finite variable logic, model checking, model theory for restricted classes of finite structures, and spatial databases. The chapters all include extensive bibliographies facilitating deeper exploration of the literature and further research.
During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.
This is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference.
I will report on joint work with G. Cherlin on the quasi-finite axiomatizability of smoothly approximable structures, and on finite structures with few types. Let L be a finite language, k an integer, and C(L, k) be the class of finite L-structures with at most k 5-types, The large members of C(L, k) with no nontrivial 0-definable equivalence relation are known to be bi-interpretable with (finite unions of) certain classical geometries. (Work of Cherlin-Lachlan, Kantor-Liebeck-Macpherson.) A dimension of an L-structure M is the dimension of a geometry interpretable in M, such that every automorphism of the geometry lifts to an automorphism of M (with some other conditions.) C(L, k) can be naturally (and effectively) divided into a finite number of families C
ϕ
(L, k). Within each C
ϕ
, a finite number of dimensions is identified; every first order statement is equivalent to a Boolean combination of statements asserting that a given dimension is finite (and fixed). In particular, the isomorphism type of a structure in F
i
is determined by its dimensions. These dimensions can be varied essentially independently. This generalizes Lachlan’s theory of shrinking and stretching homogeneous structures for a finite relational language. The proof involves methods of stability theory (geometries, orthogonality, modularity, stable groups) applied in this unstable context.
This article introduces some of the basic concepts and results from model theory, starting from scratch. The topics covered are be tailored to the model theory of fields and later articles. I will be using algebraically
We consider a difference field (K,σ) such that finite-dimensional definable sets over K can be compared in size, or measured. Let k be the fixed field of the automorphism σ. We show that curves of genus 1 defined over k are approximately the size of the affine line over k, an ‘approximative version’ of the Riemann hypothesis for curves of genus 1.
We outline the structure theory for infinite structures which are smooth limits of finite structures, or equivalently for sufficiently large finite permutation groups with a bounded number of orbits on 4-tuples. The primitive case is treated explicitly in [11] assuming a bound on orbits on 5-tuples, and modifications needed to work with a bound on 4-tuples are indicated in [15]. This theory is an extension of the theory of ℵo-categorical ℵo-stable structures. The main technical innovations at this level of generality are due to Hrushovski; some of them are useful in other semistable contexts.
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 Cherlin and Hrushovski [6]. See [5] for an account of this.
The purpose of these notes is to summarize the theory of stable homogeneous structures. The study of such very special structures provided the context in which the notion of smoothly approximated structures first arose. Cherlin’s article outlines the current state of the theory of smoothly approximated structures, a theory which seems to be rapidly approaching completion.
We attempt to formulate issues around modularity and Zilber’s trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9].
The main new result given here is the classification of the countable homogeneous directed graphs, carried out in chapters 5-8. It has long been known that there are 2(N0) such graphs. The classification of homogeneous n-tournaments given in chapters 23 serves to illustrate the methods and is occasionally useful in the latter half of the monograph. Chapter 4 gives a new proof of the corresponding theorem of Lachlan and Woodrow for symmetric graphs, in a manner quite similar to that used here in the directed case. An appendix points out some examples of homogeneous structures in richer languages which one would probably want to consider more closely before attempting to carry out arguments of a similar type in richer binary languages.
Acknowledgments The research undertaken in Chapter 6 was performed jointly by myself and Mark Ryten. My contribution to the work was throughout: in the formation of the initial strategy, in the development of the methods, in the exposition of the proofs, and in their correction and fine-tuning. Many thanks to my PhD-supervisor Dugald Macpherson for his guidance, assis- tance, and support. Thanks also to the University of Leeds and especially the sta,of the School of Mathematics for providing me with the facilities and support to undertake this re- search. Thanks to Agatha Walczak-Typke for her help with LaTeX. I’d like in particular to thank Jessica Elwes, Colin and Jessica Russell, and Haruka Okura, as well as my family and friends in general who have been a constant source of support and inspiration over the last few years. Three quarters of this research was funded by the EPSRC. 3 Abstract The primary objects of study in thesis are asymptotic classes: classes of finite structures in a fixed first order language, where the definable sets have good asymp- totic behaviour as the structures become,large. We will also consider measurable structures: infinite structures equipped with a finitary counting measure. Measur- able structures form a more general setting for the study of asymptotic classes.
Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.
2000 Mathematical Subject Classification: 03C45.
A classification is given of primitive K0-categorical structures which are smoothly approximated by a chain of finite homogeneous substructures. The proof uses the classification of finite simple groups and some representation theory. The main theorems give information about a class of structures more general than the X0-categorical, co-stable structures examined by Cherlin, Harrington, and Lachlan.
LetL be a finite relational language andH(L) denote the class of all countable structures which are stable and homogeneous forL in the sense of Fraissé. By convention countable includes finite and any finite structure is stable. A rank functionr :H(L) →ω is introduced and also a notion of dimension for structures inH(L). A canonical way of shrinking structures is defined which reduces their dimensions. The main result of the paper is that
anyM ∈H(L) can be shrunk toM′ ∈H(L),M′ ⊆M, such that |M′| is bounded in terms ofr(M), and the isomorphism type ofM overM′ is uniquely determined by the dimensions ofM. Forr<ω we deduce thatH(L, r), the class of allM ∈H(L) withr(M)≦r, is the union of a finite number of classes within each of which the isomorphism type of a structure is completely determined
by its dimensions.
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 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.
We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy .Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle : for no definable set A with more than one element can A
2 definably embed into A.
In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs ( d , μ ) ∈ ω × Q >0 so that in any finite field F and for any ā ∈ F m the size | ø ( F n ,ā)| is “approximately” μ | F | d . Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.
Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer coefficients. Moreover, the shifted periodic polynomial function, where $q^n$ is formally replaced with $q^n + 1$, is shown to have non-negative coefficients.
Multidimensional asymptotic classes
- S Anscombe
- H D Macpherson
- C Steinhorn
- D Wolf
Model theory of multidimensional asymptotic classes
- D Wolf
A re-iteration of Finite structures with few types, Chapters 1-3, Unpublished manuscript
- C Hill
- C Smart
Model theory of finite and pseudofinite rings, PhD thesis
- R I Bello Aguirre
This is an unabridged republication of the 1974 third revised printing of the work originally published by the North-Holland Publishing Company in 1969
- J L Bell
- A Slomson
Examples of -categorical structures, Automorphisms of First-Order Structures
- D M Evans
- R Kaye
- H D Macpherson