On GLn(B) Where B is a Boolean Ring

1975 ◽  
Vol 18 (2) ◽  
pp. 209-215 ◽  
Author(s):  
H. Gonshor
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Stone Space ◽  
Boolean Ring ◽  
The Subject ◽  
Topology Model

The aim of this paper is to generalize the main results of [1] to GLn(B) by means of proofs which are more conceptual and less computational. In addition, by means of the Stone space we will obtain results which are new even for the case n = 2. Finally we shall make some remarks of a categorical nature.The author is especially interested in the subject because of the overlap here of many areas of mathematics. Concepts from topology, model theory, and category theory are all relevant.

Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

2003 ◽  
Vol 9 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Theoretic Model ◽  
Topos Theory ◽  
Sheaf Theory ◽  
Special Cases ◽  
Algebraic Spaces ◽  
The Subject ◽  
Near Future ◽  
Modern Mathematics

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

scholarly journals A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

2018 ◽  
Vol 83 (1) ◽  
pp. 326-348 ◽  
Author(s):  
RUSSELL MILLER ◽  
BJORN POONEN ◽  
HANS SCHOUTENS ◽  
ALEXANDRA SHLAPENTOKH
Keyword(s):  
Model Theory ◽  
Open Problem ◽  
Category Theory ◽  
Computable Model ◽  
Computable Model Theory ◽  
Arbitrary Characteristic ◽  
Faithful Functor ◽  
New Construction ◽  
Image Position ◽  
Countable Structure

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

Groups and Monoids of Regular Graphs (And of Graphs with Bounded Degrees)

1973 ◽  
Vol 25 (2) ◽  
pp. 239-251 ◽  
Author(s):  
Pavol Hell ◽  
Jaroslav Nešetřil
Keyword(s):  
Category Theory ◽  
Main Question ◽  
Regular Graphs ◽  
Group Of Automorphisms ◽  
Automorphisms Of Graphs ◽  
Usual Manner ◽  
The Subject ◽  
Groups Of Automorphisms

A graph X is a set V(X) (the vertices of X) with a system E(X) of 2-element subsets of V(X) (the edges of X). Let X, Y be graphs and f : V(X) → V(Y) a mapping; then/ is called a homomorphism of X into F if [f(x),f(y)] ∈ E(Y) whenever [x,y] ∈ E(X). Endomorphisms, isomorphisms and automorphisms are defined in the usual manner.Much work has been done on the subject of representing groups as groups of automorphisms of graphs (i.e., given a group G, to find a graph X such that the group of automorphisms of X is isomorphic to G). Recently, this was related to category theory, the main question being as to whether every monoid (i.e., semigroup with 1) can be represented as the monoid of endomorphisms of some graph in a given category of graphs.

STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES

2016 ◽  
Vol 81 (3) ◽  
pp. 1069-1086
Author(s):  
CHARLES C. PINTER
Keyword(s):  
Boolean Algebra ◽  
Model Theory ◽  
Representation Theorem ◽  
Stone Space ◽  
Equivalence Relations ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
Abstract Model Theory ◽  
Image Position

AbstractThe Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.

scholarly journals Intelligently deciphering unintelligible designs: algorithmic algebraic model checking in systems biology

2009 ◽  
Vol 6 (36) ◽  
pp. 575-597 ◽  
Author(s):  
Bud Mishra
Keyword(s):  
Systems Biology ◽  
Model Checking ◽  
Model Theory ◽  
Historical Context ◽  
Algebraic Model ◽  
The Subject ◽  
And Algebra

Systems biology, as a subject, has captured the imagination of both biologists and systems scientists alike. But what is it? This review provides one researcher's somewhat idiosyncratic view of the subject, but also aims to persuade young scientists to examine the possible evolution of this subject in a rich historical context. In particular, one may wish to read this review to envision a subject built out of a consilience of many interesting concepts from systems sciences, logic and model theory, and algebra, culminating in novel tools, techniques and theories that can reveal deep principles in biology—seen beyond mere observations. A particular focus in this review is on approaches embedded in an embryonic program, dubbed ‘algorithmic algebraic model checking’, and its powers and limitations.

scholarly journals SPACES OF TYPES IN POSITIVE MODEL THEORY

2019 ◽  
Vol 84 (02) ◽  
pp. 833-848
Author(s):  
LEVON HAYKAZYAN
Keyword(s):  
Model Theory ◽  
Stone Space ◽  
Distributive Lattices ◽  
Stone Duality ◽  
Order Model ◽  
First Order ◽  
Countable Models

AbstractWe introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

Degree Spectra of Relations on Computable Structures

2000 ◽  
Vol 6 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Model Theory ◽  
Large Body ◽  
Point Of View ◽  
Linear Ordering ◽  
Computable Model ◽  
Computable Model Theory ◽  
Degree Spectra ◽  
Successor Relation ◽  
The Subject ◽  
Cover Material

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. (A valuable reference is the handbook [7]. In particular, the introduction and the articles by Ershov and Goncharov and by Harizanov give useful overviews, while the articles by Ash and by Goncharov cover material related to the topic of this communication.)Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable (c. e.) degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough.

Properties of first-order logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Model Theory ◽  
Propositional Logic ◽  
Final Section ◽  
Order Logic ◽  
Countable Model ◽  
First Order Logic ◽  
First Order ◽  
Beth Definability ◽  
The Subject ◽  
Infinite Model

We show that first-order logic, like propositional logic, has both completeness and compactness. We prove a countable version of these theorems in Section 4.1. We further show that these two properties have many useful consequences for first-order logic. For example, compactness implies that if a set of first-order sentences has an infinite model, then it has arbitrarily large infinite models. To fully understand completeness, compactness, and their consequences we must understand the nature of infinite numbers. In Section 4.2, we return to our discussion of infinite numbers that we left in Section 2.5. This digression allows us to properly state and prove completeness and compactness along with the Upward and Downward Löwenhiem–Skolem theorems. These are the four central theorems of first-order logic referred to in the title of Section 4.3. We discuss consequences of these theorems in Sections 4.4–4.6. These consequences include amalgamation theorems, preservation theorems, and the Beth Definability theorem. Each of the properties studied in this chapter restrict the language of first-order logic. First-order logic is, in some sense, weak. There are many concepts that cannot be expressed in this language. For example, whereas first-order logic can express “there exist n elements” for any finite n, it cannot express “there exist countably many elements.” Any sentence having a countable model necessarily has uncountable models. As we previously mentioned, this follows from compactness. In the final section of this chapter, using graphs as an illustration, we discuss the limitations of first-order logic. Ironically, the weakness of first-order logic makes it the fruitful logic that it is. The properties discussed in this chapter, and the limitations that follow from them, make possible the subject of model theory. All formulas in this chapter are first-order unless stated otherwise. Many of the properties of first-order logic, including completeness and compactness, are consequences of the following fact: Every model has a theory and every theory has a model. Recall that a set of sentences is a “theory” if it is consistent (i.e. if we cannot derive a contradiction). “Every theory has a model” means that if a set of sentences is consistent, then it is satisfiable.

Gregory Maxwell Kelly 1930 - 2007

2010 ◽  
Vol 21 (2) ◽  
pp. 237
Author(s):  
Ross Street
Keyword(s):  
Category Theory ◽  
New South ◽  
Professor Emeritus ◽  
Senior Lecturer ◽  
University Of Cambridge ◽  
South Wales ◽  
Pure Mathematics ◽  
Mathematical Discipline ◽  
The Subject ◽  
The University

Gregory Maxwell (?Max') Kelly (1930?2007) was educated at the University of Sydney (BSc 1951 with First Class Honours, University Medal for Mathematics, Barker Prize, and James King of Irrawang Travelling Scholarship) and the University of Cambridge (BA 1953 with First Class Honours and two Wright's Prizes; Rayleigh Prize, 1955; PhD 1957). He returned to Australia as Lecturer in Pure Mathematics at the University of Sydney in 1957, became Senior Lecturer in 1961 and Reader in 1965. He was appointed Professor of Pure Mathematics first at the University of New South Wales in 1967 then at the University of Sydney in 1973, becoming Professor Emeritus in 1994. He introduced the mathematical discipline of category theory to Australia and continued active and influential research in the subject until the day of his death.

scholarly journals The Boolean ring universal over a meet semilattice

1977 ◽  
Vol 23 (4) ◽  
pp. 402-415 ◽  
Author(s):  
Elliot Evans
Keyword(s):  
Vector Space ◽  
Congruence Lattice ◽  
Linear Independence ◽  
Stone Space ◽  
Ideal Lattice ◽  
Boolean Ring ◽  
The Ideal

AbstractThe Boolean ring B{M} universal over a meet semilattice M is examined. It is the vector space over the two element field Z2 with base M\{0}. The Z2 linear independence of a meet subsemilattice of a Boolean ring is characterized in order theoretic terms and some ramifications of this on B[M] are considered. The space (ℱpM) of proper filters of M is shown homeomorphic to the Stone space S(B[M]) of B[M] if M has no least element, with ℱP(M) ∪ {M} and S(B[M]) homeomorphic otherwise. The congruence lattice θ(M) of M is compared to the ideal lattice ℱ(B [M]) of B [M] with best results coming if M is a tree with zero when θ (M) = ℐ (B [M]).

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