Harold Simmons

A Framework for Topology

Publisher Summary For each topological space S, the topology OS of S (the family of open sets of S) is a lattice, and many topological properties of S are describable as lattice theoretic properties of OS. If S is a reasonably separated space then S is completely determined by OS. A closer study should be made of the lattice theoretic properties of topologies and of how these properties control the properties of the parent space. This chapter discusses one such theory, “frame theory.” Much of frame theory is concerned with the properties of nuclei. The first result of frame theory is a characterization of the kernels of frame morphisms. There are frames that are not topologies and each frame has a canonically associated space. The construction of this space is analogous to the construction of the stone space of a distributive lattice.

Near-discreteness of modules and spaces as measured by Gabriel and Cantor

Abstract The Gabriel dimension analysis of a module category and the Cantor-Bendixson rank analysis of a topological space have a common generalization via a lifting to certain lattices. In many instances the Gabriel derivative and the Cantor-Bendixson derivative coincide.

An Introduction to Category Theory by Harold Simmons

Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.

Existentially Closed Models of Basic Number Theory

This paper is concerned with a subtheory B of (first order) peano number theory P which is strong enough to contain many non-trivial number theoretic facts. We call this theory B basic number theory. Our aim is to give a complete description of the spectrum of (countable) models of B. Of course we will not achieve this aim here, but the results we give and the methods we use do give us some idea of the overall picture.

Regularity, Fitness, and the Block Structure of Frames

I study the point-free and point-sensitive aspects of fitness, subfitness, and the difference between these two.

Proof theory: a selection of papers from the Leeds Proof Theory Programme 1990

Preface Programme of lectures 1. Basic proof theory S. Wainer and L. Wallen 2. A short course in ordinal analysis W. Pohlers 3. Proofs as programs H. Schwichtenberg 4. A simplified version of local predicativity W. Buchholz 5. A note on bootstrapping intuitionistic bounded arithmetic S. Buss 6. Termination orderings and complexity characterisations E. Cichon 7. Logics for termination and correctness of functional programs, II. Logics of strength PRA S. Feferman 8. Reflecting the semantics of reflected proof D. Howe 9. Fragments of Kripke-Platek set theory with infinity M. Rathjen 10. Provable computable selection functions on abstract structures J. Tucker and J. Zucker.

Compact representations—The lattice theory of compact ringed spaces

Abstract By extracting the lattice theoretic content we see that the universal compact representation of a ring is canonically determined by the regular core of its lattice of two-sided ideals.

Large and Small Existentially Closed Structures

The generic structures introduced by Abraham Robinson are now well established in model theory. Anyone who works with these structures soon begins to feel that, in some sense, the infinite generic structures are large and the finite generic structures are small. Here ‘large’ and ‘small’ do not refer to the cardinality of the structures but are used in the way we describe saturated structures as large and atomic structures as small. In this paper I isolate what I consider to be the large and small existentially closed (e.c.) structures and attempt to determine their role in the class of all e.c. structures. In the usual context of model theory we are concerned with elementary embeddings and all formulas. E.c. structures are concerned with all embeddings and ∃ 1 -formulas. Thus we need to look at ∃ 1 -analogues of saturated and atomic structures. These ∃ 1 -saturated structures have been around for some time; they are just the existentially universal structures. The corresponding ∃ 1 -atomic structures are not new here (they appear in [8]) but I believe that this paper will add much to the understanding of them. The bulk of this paper is in §§2 and 4. §2 deals with ∃ 1 -atomic structures, and §4 is concerned with the ∃ 1 -analogues of several results in Vaughts paper [12]. A similar program has been carried out by Pouzet in [6], [7], [8]; however, there the significance of e.c. and e.c. structures is not realized and consequently some of the simplicity of the situation is lost.

The semiring of topologizing filters of a ring

The semiring of topologizing filters of a ring has a more amenable description in terms of increasing functions on the lattice of (one sided) ideals.

Each regular number structure is biregular

Roughly speaking we show that for certain number structures ℋ, ℬ with ℬ ⊆ ℋ, if ℬ is bounded above in ℋ then ℬ is bounded below in ℋ.