Categorical Structures And Their Applications: Proceedings Of The North-west European Category Seminar, Berlin, Germany 28 - 29 March 2003

CATEGORICAL STRUCTURES AND THEIR APPLICATIONS

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Proceedings of the North-West European Category Seminar

CATEGORICAL STRUCTURES AND THEIR APPLICATIONS Berlin, Germany

28 - 29 March 2003

edited by W.Gahler & G.Preuss Freie Universitat Berlin, Germany

'World Scientific NEW J E R S E Y • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: UK office:

27 Warren Street, Suite 401-402, Hackensack, NJ 07601 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

CATEGORICAL STRUCTURES AND THEIR APPLICATIONS Proceedings of the North-West European Category Seminar Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-053-X

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Dedicated to Horst Herrlich on the occasion of his 65th birthday

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Horst Herrlich at work

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Contents Preface

xiii

Introduction

xv

Opening Ceremony of the North-West European Category Seminar 2003

xvii

On Boole's Booleanness B. Banaschewski

1

Homeomorphically Closed Nearness Spaces

H. L. Bentley and John W. Carlson

13

The Tensor Product of Orthomodular Posets

Reinhard Borger

29

On a Weak Form of the Blumberg Property Wing-Sum Cheung, Yu-Ting Lin, George E. Strecker and Shiojenn Tseng

41

Saturated Collections of Metrics E. Colebunders, R. Lowen and M. Sioen

51

A Uniform Space Proof of a Metrisation Theorem

P. J. Collins

67

Classification of Closure Operators for Categories of Topological Spaces Dikran Dikranjan, Walter Tholen and Stephen Watson . . .

69

Topological Structures in Logics Werner Gahler

99

The Structure of Affine Algebraic Sets Eraldo Giuli

113

A Twisted Triple Category of Track Commutative Cubes K. A. Hardie and K. H. Kamps

IX

121

xx

CONTENTS Productivity of Coreflective Classes in Some Topological Structures Miroslav Husek

143

A Characterization of Co-Retracts of Functional Structures Roland Kaschek

157

Maximal (Sequentially) Compact Topologies Hans-Peter A. Kiinzi and Dominic van der Zypen

173

Supertopologies as Starting Points for Generalized Continuity Structures Dieter Leseberg

189

Introducing Lagois Correspondences Austin Melton

207

On (B,B)-Projectivity Helga Oltmanns, Valdis Laan, Ulrich Knauer and Mati Kilp

219

On Coalgebras which are Algebras Hans-E. Porst and Christian Dzierzon

227

A Hyperspace Completion for Semiuniform Convergence Spaces and Related Hyperspace Structures Gerhard Preuss

237

Convex Effect Algebras and Partially Ordered Positively Convex Modules D. Pumpliin and H. Rohrl

251

Fibrewise Sobriety Giinther Richter and Alexander Vauth

265

Two Applications of Elementary Submodels to Partitions of Topological Spaces J. Schroder and Steve Watson

285

L-Valued Categories: Generalities and Examples Related to Algebra and Topology Alexander P. Sostak

291

CONTENTS

xi

Horst Herrlich's Publications

313

Bernhard Banaschewski: Two Poems

333

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Preface The North-West European Category Seminar 2003 in Berlin was devoted to Horst Herrlich on the occasion of his 65th birthday and his retirement. The organizers of this seminar, Werner Gahler und Gerhard Preufi, have decided to publish a proceedings volume with additional contributions of invited colleagues from everywhere and to dedicate this volume to Horst Herrlich, one of the leading category theorists of the world. During the preparation of this book Bernhard Banaschewski sent two poems for Horst Herrlich to the editors, which will be presented at the end of this edition in order to demonstrate that mathematicians are often humorous people. A high value is set on the applications of category theory according to the varied interests of Horst Herrlich. They include the following areas: 1) Algebra (Dierzon, Kilp, Knauer, Laan, Oltmanns, Porst, Pumplun, Rohrl). 2) Computer Science (Kaschek, Melton). 3) Fuzzy Structures and Logic (Gahler, Sostak). 4) Order Theory (Banaschewski, Borger). 5) Topology: a) Homotopy Theory (Hardie, Kamps). b) Fibrewise Topology (Richter, Vauth). c) Special Topological Constructs (Bentley, Carlson, Cheung, Colebunders, Collins, Dikranjan, Giuli, Husek, Kiinzi, Leseberg, Lin, Lowen, PreuC, Tholen, Schroder, Sioen, Strecker, Tseng, Watson, van der Zypen). We are very grateful to Mrs. Barbara Wengel for collecting the papers and to Mr. Attila Kimpan for preparing them according to the rules of the publisher. Furthermore, we would like to thank Professor Jorg Raasch for taking a picture of Horst Herrlich at work. Last but not least we are indebted to the scientific publisher, Dr. Jitan Lu, from World Scientific for quick publication.

Gerhard Preufi

Werner Gahler

Xlll

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Introduction As is well-known category theory was created by S. Eilenberg and S. Mac Lane in 1945. For a considerable time after this event category theory served mainly as a suitable language for formulations in the framework of homological algebra and algebraic topology. The next decisive notion, namely that of an adjoint functor, was introduced in 1958 by D. M. Kan, though the closely related concept of a universal arrow has been studied by P. Samuel ten years earlier, at least in the framework of Bourbaki's structures. The equivalence of the concepts of adjoint functors and universal arrows (now defined in the realm of category theory) was proved explicitely by J. Sonner in 1964. Also in the sixties of the 20th century the study of applications of adjoint situations were intensified, e.g. reflections and coreflections (P. Freyd, J. Isbell, H. Kennison, H. Herrlich and others) or cartesian closedness (E. Binz, M. Katetov, N. Steenrod and others). In 1971 the field of categorical topology has been established definitively by H. Herrlich and O. Wyler. Its aim is the study of various categories being interesting for topologists, and the relations between them by means of categorical methods. The so-called topological constructs take up a considerable part of this volume. Approximately at the same time theoretical computer scientists became interested in category theory, e.g. in the realm of automata theory (J. A. Goguen, M. Arbib and E. G. Manes, H. Ehrig and M. Pfender). Another discipline in which categorical methods are used is fuzzy theory introduced by A. Zadeh in 1965. Fuzzy theory is related to control theory and to neural networks and provides a lot of contributions to category theory, to different types of logic, and to different types of topological structures (B. Schweizer, A. Sklar, E. Colebunders, R. Lowen, D. Eklund, W. Gahler, U. Hohle, A. P. Sostak and others). It is not the intention of this introduction to give the reader a detailed idea of the historical development of category theory and its applications but much more this book should be a useful guide to learn a lot about the latest development of categorical methods in various fields of its applications.

xv

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Opening Ceremony of the North-West European Category Seminar 2003

Dear colleagues, dear Horst, The organizers of the North-West European Category Seminar 2003, Werner Gahler and I, welcome you very cordially in Berlin. This year, the seminar is devoted to our colleague (and friend of many of us) Horst Herrlich on the occasion of his 65th birthday and his retirement. Let me use this opportunity to give a short survey of his mathematical career. In 1956 (nineteen years after his birth in Berlin) he studied mathematics and biology mainly at the Free University with the goal to become a school teacher. Indeed, he was a school teacher from 1961-1963, but already in 1962 he got his PhD from the Free University in the age of 25 for his thesis "Ordnungsfahigkeit topologischer Raume", and in 1963 he became an assistant in Mathematics at the same university. The students liked him because he was always happy and helped them in their mathematical problems. I know this very well, since I was one of these students, and I am very grateful for his supervision. In 1965 he finished his habilitation thesis on "£-kompakte Raume" and became a "Privatdozent" in 1966 and a professor in 1969. One of his first courses was entitled "Topologische Reflexionen und Coreflexionen" and appeared in 1968 as a book in the Springer Lecture Notes in Mathematics. It is very famous because it is fundamental for the so-called Categorical Topology which started in 1971, mainly by Horst Herrlich and Oswald Wyler. Also at the beginning of his career Horst was invited to be a Visiting Assistant Professor at the University of Florida in Gainesville from 1966 to 1967 and from 1969 to 1970. There he met George Strecker and a life-long friendship started and culminated 1973 in the appearance of the joint book "Category Theory" which is and was a must for every category theorist. This book has been enlarged in 1990 (together with J. Adamek) and appeared under the title "Abstract and Concrete Categories". In 1970 Horst went to Bielefeld and 1971 to Bremen where he retired last year as a full professor in Mathematics. During his time in Berlin 18 papers appeared in well-reputed journals and two papers in Conference Proceedings. This period is also characterized by close connections to the topological school in the Netherlands. I mention here: J. de Groot, J. van der Slot, E. Wattel and last not least George Strecker. xvii

xviii

OPENING CEREMONY

In the seventies Horst published 42 papers, some of them with coworkers. Among them are: J. Adamek, B. Banaschewski, H.L. Bentley, W.W. Comfort, V. Kannan, M. Rajagopalan, W.A. Robertson, G. Salicrup, G. Strecker, and R. Vazquez. In this period he developped not only the fundamentals of Categorical Topology (including topological functors, initial and final completions, and cartesian closedness of topological constructs) but also a new foundation of General Topology by axiomatizing the intuitive concept of nearness between an arbitrary collection of sets. The resulting nearness spaces are essential for the extension theory of topological spaces. His completion of nearness spaces is known today as Herrlich completion. Furthermore, he recognized the importance of factorization structures in Category Theory and developped e.g. dispersed and light factorization structures. In the eighties 37 papers appeared together with a book on topology consisting of three volumes where the last volume is entirely devoted to nearness spaces. His book demonstrates that H2 (= Horst Herrlich) is also an excellent teacher. Since the development of Categorical Topology was very rapid, H2 published a paper entitled "Categorical Topology 19711981" with merely 700 references and several open problems. Futher, he invented Universal Topology - a counterpart of Universal Algebra. For topological constructs, Horst established another convenient property besides cartesian closedness, namely heriditariness, i.e. the existence of one-point extensions. In Category Theory he made precise the difference between algebra and topology. Additionally, he studied Galois connections in a joint paper with M. Husek in 1984, and then in a pure categorical manner in 1990. In the nineties the contacts to friends in Belgium, in particular to R. Lowen and E. Lowen-Colebunders, were intensified, and some joint papers appeared, e.g. on convergence or on improvements of topological constructions. Approximately from the middle of the nineties up to the presence H2 has also worked in the interactions of axiomatic set theory and general topology and got highly interesting results, e.g. on the axiom of choice and the Boolean prime ideal theorem. In the same period several historical overviews on "Categorical Topology", on "Uniform, proximal and nearness concepts", and on several themes in connection with Felix Hausdorff's collected works (together with coworkers) appeared. Summarizing Horst has created an impressive and stimulating work of 165 papers1, six books (mentioned before), and a mathematical biography on G. Salicrup. Furthermore, he has edited 14 conference proceedings (with various coeditors). l

ln the meantime there are 170 papers (cf. p. 313 ff).

OPENING

CEREMONY

xix

It's time for finishing this overview and saying: Hearty congratulations to your birthday, dear Horst! Remain in good health and have good ideas in the future for the benefit of all of us!

Gerhard Preufi

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Upper row (from left to the right): M. HuSek, G. Gutierres, E. Colebunders, R. Lowen, Ch. Dzierzon, H. Herrlich, G. Preuss, H.-E. Porst, J. Schroder, R. Brandt, G. Richter Lower row (from left to the right): H. Kamps, M. Sioen, T. Kubiak, Ch. Schubert, U. Knauer, D. Leseberg, R. Borger, W. Gahler, T.-U. Jandek

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ON BOOLE'S BOOLEANNESS

B. BANASCHEWSKI Department of Mathematics & Statistics McMaster University Hamilton, Ontario, Canada L8S 4K1 E-mail: [email protected] This paper provides an analysis of the formalism presented in Chapters II and III of Boole's "Laws of Thought", showing that it amounts to describing a particular kind of pseudocomplemented meet-semilattice, very far from Boolean algebras. In addition, several identities and identical implications are considered which do turn these semilattices into Boolean algebras. Mathematics Subject Classifications (2000): 06A12,03G99,03G05 Keywords: Boolean algebras,pseudocomplemented meet-semilattices,the laws of excluded middle and double negation as identities

As is well known, George Boole's seminal work in symbolic logic did not quite introduce the concept of Boolean algebra as now understood but rather a more rudimentary system of which it is a streamlined and more evolved version: "originated by Boole, extended by Schroder, and perfected by Whitehead" (Sheffer[7]). This article, going back to the beginnings, presents an analysis of the precise algebraic nature of Boole's own original system as defined by his explicitly given operations and laws. We take this to be presented in its definitive form in Boole[2], given the fact that Boole himself viewed this as the more mature version of its forerunner Boolefl]. Below, all reference to Boole[2] are given by chapter and section numbers. Special note should be taken of the above emphasis on explicit: it is true that, in several instances, Boole allows himself forms of reasoning not covered by his explicit stipulations, and it could perhaps be argued that the latter were not intended as the definition of an algebraic system in our precise sense, so that taking him all too literally might run the risk of misconstruing the text - but an author's intentions are one thing and explicit formulations another, and we believe the latter well worth 1

2

BANASCHEWSKI

analyzing, all other considerations notwithstanding. It turns out, and may come as a bit of a surprise, that Boole's specifications describe algebraic systems that are actually quite far from Boolean algebras proper (Proposition 1) although they do lead to the latter by the addition of the axiom expressing either the Law of Double Negation or the Law of the Excluded Middle (Proposition 2), neither of which, somewhat curiously, appears among the explicitly listed laws. We augment this result by discussing some further conditions equivalent to the Law of the Excluded Middle in this context (Proposition 3). Boole defines what amounts to an algebra of classes by introducing one binary operation, multiplication (xy), two partial binary operations, addition (x+y) and subtraction (x — y), and two distinguished elements, 0 and 1, motivated, respectively, by class intersection, union of disjoint classes, the exclusion of members of one class from a bigger class, the empty class and the universal class. The laws explicitly stated concerning multiplication say, in modern language, that we have an idempotent commutative monoid with zero, 1 being the unit and 0 the zero (116, 119, III 13). It should be explained that, although the familiar expression of associativity,

x(yz) = (xy}z does not actually occur in the text, it may nonetheless be directly inferred from the statement (II7) that zxy and xyz represent the same class since the natural reading of this, involving consistent bracketing,

z(xy) = x(yz)

or

(zx)y = (xy)z ,

together with commutativity yields the desired identity. As to the (really not relevant) inconsistent bracketing,

z(xy) = (xy)z

or

(zx)y - x(zy)

we observe that the first produces nothing new, while the second again leads to associativity. Further, expressing Boole's stipulations for the domains of the two partial operations in algebraic terms, x + y is defined for all x, y such that xy = 0, and

x - y for all x, y such that xy = y, while the laws stated for these are (II11 and II13)

x+y =y+x

(1)

BOOLE'S BOOLEANNESS

z(x + y) = zx + zy,

z(x - y) = zx - zy

if x = y + z then x — z — y .

3

(2)

(3)

What is missing here is a law that governs the distinguished elements with respect to + and —, evidently needed since 10 = 0. Obviously, the most natural way to fill this gap is to pose 1 = 0 + 1,

(4)

immediately yielding

x = 0+x for arbitrary x by (2), which in turn has the two consequences

x - 0 = x,

x —x = 0

(5)

by (3). Note that Boole certainly uses the second identity of (5), but without any formal justification. A comment concerning the operations + and — may be in order here. As to +, Boole makes it unambiguously clear in several places that the motivation is given by the union of disjoint classes, and on that basis it must be viewed as a partial operation in the way described here. On the other hand, at a later stage, he does permit himself to write expressions which no longer satisfy his original restriction on + and provides some discussion as to the meaning of taking this liberty. Similar remarks apply to the operation —. Now, it clearly is an important question how Boole's freewheeling extension of the symbolism, ultimately some kind of passage from a partial to a total algebra, should properly be accounted for, but this is an issue quite different from the concern here. We note that this question is dealt with by Hailperin[4] who places it in the setting of semiprime commutative rings, with unit and torsionfree additive group, where class algebra is recaptured in the form of the corresponding algebra of idempotents. This may be an attractive approach to the problem involved, but in the final analysis it takes it as settled that the notion Boole introduces in Chapters II and III is Boolean algebra proper; our concern, on the other hand, is the prior question to what extent this is actually the case, provided one specifically focuses on what is explicitly stated. Going somewhat beyond Boole himself, it might be worth adding here, again concerning +, that most of Boole's successors (the exception being Venn) decided to deviate from him and to take + as a total operation; indeed, it seems this step was seen as the great advance over Boole's system,

4

BANASCHEWSKI

as indicated by the extensive discussion of this point by Schroder [7], more recently reiterated by Kneale[6]. There is one further detail that requires comment. After introducing his operations and laws, Boole compares his algebra of logic with the usual algebra of numbers (1114,15). Noting that the only numbers for which z2 = x are 0 and 1, he envisages an algebra of number symbols restricted to taking on only these two values and then claims that this will be formally the same as his algebra of logic (II15). This is a dubious statement which we feel free to ignore for our purposes since it can serve neither as an encapsulation of the preceding presentation of operations and laws, nor as a meaningful augmentation of the principles laid down there: for instance, if x and y only range over {0,1} then xy = 0 implies x = 0 o r y = 0 - a n implication Boole surely would not want to claim for his algebra of logic. For similar criticism of II 15 we refer to Kneale[4]. In the following, then, we regard a partial algebra with the above described operations and laws as the embodiment of Boole's explicit formulations in Chapters II and III, and call it a Boolean system. Our first aim is to relate this to more familiar notions. To begin with, note that any Boolean system is a meet-semilattice in which x < y iff x = xy and x Ay = xy, with 0 and 1 as the bottom and top, that is, zero and unit in the partial order sense. In any meet-semilattice with 0, x and y are called disjoint if x A y = 0; further, whenever elements x and y have a join (= least upper bound) this will be denoted xVy&s usual, and it will be called distributive if z A (x V t/) = (z A x) V (z A y) for all z; finally, we say a meet-semilattice with 0 has distributive disjoint joins provided any pair of disjoint elements has a join and this is distributive. Also, recall that a meet-semilattice is called pseudocomplemented if it has a zero 0 and any element x has a pseudocomplement, that is, there is an element x*, necessarily unique, such that x A y = 0 iff y < x*, for all y. Now we have Proposition 1. Any Boolean system B is pseudocomplemented such that x* = 1 — x for any x € B, and for any disjoint x,y 6 B, x V y — x + y. Conversely, any pseudocomplemented meet-semilattice with unit and distributive disjoint joins is a Boolean system with xy = x/\y, x + y = x\/y for any disjoint x and y, and x — y = x A y* for any x > y.

Proof. If xy = 0 in B then y(l — x) = y — yx = y and hence y < I — x-

BOOLE'S BOOLEANNESS

5

conversely, this implies xy = xy(l —x)= y(x — x2) = y(x — x) — 0

since x — x = 0 by (5). This shows x* = I — x. Further, for any disjoint x and y

x(x + y)=x,

y(x + y) = y

so that x, y < x + y; on the other hand, if x, y < z then x = xz and y = yz, and therefore

x + y = xz + yz = (x + y)z , showing x + y < z. It follows that x + y = x V y. For the converse, define xy = x A y for all x, y, x + y = x V y for any disjoint x and y, and x — y = x A y* for any x > y. Then, clearly, (1) and the first identity of (2) hold. Further, for any x > y, z(x — y} = z/\x/\y* while zx — zy = z A x A (z A y)*

and since y* < (z A y)* and z A (z A y)* < y* by the properties of pseudocomplements, these two are equal, which proves the second identity of (2). Next, if x = y + z = y V z for disjoint y and z, then x-z = (yVz)/\z*=yAz*=y, the last step since y A z = 0, showing (3). Finally, (4) is obvious.

D

The second part of Proposition 1 immediately leads to the following Corollary 1. Any pseudocomplemented bounded distributive lattice, and in particular any Heyting algebra, is a Boolean system. In particular, then, any Boolean algebra is a Boolean system, but of course this corollary covers considerably more than that. In addition, it should be noted that a Boolean system need not even be a lattice , as the following example shows: For any meet-semilattice M with zero and unit 1, let B result from M by adding a new zero 0, Then B is pseudocomplemented:

6

BANASCHEWSKI

Further, any disjoint pair x, y e B has trivial join: since x A y = 0 if B iff x = 0 or y = 0 we have 1 = 01 a {y(* x Vy = > ^~ ' \x(y = 0)

and these joins are clearly distributive. Hence, by Proposition 1, B is a Boolean system, and whenever M is not a lattice B is not a lattice, either. Remark 1. It may be worth pointing out that our adoption of the condition 0+1 = 1 is crucial for the first part of Proposition 1: indeed OVl = 1+0 means 1 = 1 + 0. Given that Boole's successors considered the shift from partial to total + as a particularly significant advance, it should be emphasized that this step alone does not create Boolean algebra by any means, as the above corollary makes very clear. Thus, when Schroder[7] presented his "extended" version of Boole's algebra of logic, describing a full-fledged bounded complemented distributive lattice in current parlance, he introduced more modification of the original Boolean formalism, as laid out in II and III, than merely the totality of +. The obvious question exactly what makes a Boolean system actually a Boolean algebra, meaning: i n its standard partial order, is answered by Proposition 2. The following are equivalent for any Boolean system B: (i) B is a Boolean algebra. (ii) B satisfies the Law of Double Negation: 1 — (1 — x) = x, for all x e B. (Hi) B satisfies the Law of the Excluded Middle: x + (1 — x) = I , for all x£B. (iv) B satisfies the Extended Law of the Excluded Middle: x = xy + (x — xy) for all x, y 6 B. (v) B is a lattice such that x V y = x + y(l — x), for all x, y € B. (vi) For all x, y e B, (x - xy) + y = x + (y - xy). Proof, (i) =» (ii). Trivial. (ii) =>• (Hi). In any pseudocomplemented semilattice, (a V &)* = a* A 6*

for any distributive join that may exist, as an immediate consequence of the definition of pseudocomplement. Hence, by Proposition 1, the present

BOOLE'S BOOLEANNESS

7

hypothesis yields x + (1 - x) = x V x* = (x V x*)** = (x* A x**)* = 0* = 1, as desired. (iii) => (iv). Multiply the given condition for y by x. (iv) => (v). If x, y < z so that x = xz and y — yz then

(x + (y - xy))z = xz + (yz - xyz) = x + (y - xy) showing x + (y — xy) < z. On the other hand,

x(x + (y - xy)) = x + x(y - xy) = x by the general rules and

y(x + (y- xy)) = yx + (y - xy) = y by (iv), hence x , y < x + (y — xy) and in all then xVy = x + (y — xy). (v) =>• (vi). Immediate since y V x = x V y. (vi) =$• (v). The given condition implies

(x - xy) +xy = x(x + (y - xy)) = x, hence (iv) and consequently (v) as already shown, (v) =>• (i). We first prove distributivity: z A (x V y) = z(x + (y — xy)) = zx + (zy — zxzy) = zx V zy = (z A x) V (z A y).

Further, x V y = x + (y — xy) implies 1 = x + (1 — x) for y = 1, hence 1 — x is the complement of x, and J5 is a Boolean algebra. D Remark 2. The maps

( x , y ) >->x + y(l -x),

(x, y) t-+ x(l - y) + y

are the two obvious ways of extending + in a Boolean system to a total operation, and (vi) then says these extensions are the same iff the system is a Boolean algebra. Remark 3. The following result places the equivalence of (i) and (ii) into a considerably wider context: For any pseudocomplemented meet-semilattice S with unit, T = {x £ S | x = x**} is a Boolean algebra with 0,A, and 1 as in S, join given by (x* Ay*)*, and complement x* for x.

8

banasckewski BBB kewskiwskisANASCHEWSKI

For the first part, we only need to show that x A y G T whenever x,y € T since 0 = 0** and 1 = 1** trivially. Now x A y < x,y implies (x A y)** < x** = x and the same for y, showing (x A y}** < x A y, the non-trivial part of the identity claimed. Next, since x* A y* < x*,y* we have x, y < (x* A y*)* for any x, y e S, and if x, y < z in T then z* < x* A y* so that (a:* Ay*)* < * * * = * , showing (x* A y*)* = x V y. Further, for any x £ S, (x*)** = x* so that x* € T, and since x V x* = (x* A x**)* = 0* = 1 in T, x* is a complement of x in T. Finally, the resu Iting lattice is distributive:

z A (x V y) A ((2 A x) V (z A y))* = z A (x V y) A (2 A x)* A (z A y)* < (x V y) A x* A j/* = (x V y) A (x V ?/)* =0.

the second step because z A (z A x)* < x* since z A (z A x)* A x = 0. It follows that

z A (x V y) < ((z A x) V (x A y))** = (z A x) V (z A j/), the non-trivial part of distributivity. Now, the condition (ii) in the above proposition amounts to saying that B = {x £ B | x = x**}, making B itself the Boolean algebra considered here, showing (i). It should be added that, for pseudocomplemented distributive lattices with unit, the above result is the classical Glivenko Theorem; it is noteworthy that this still holds under the present considerable weaker hypotheses. Remark 4. Passing from Boolean systems to mere pseudocomplemented meet-semilattices with unit, the preceding discussion shows that, already in this setting, (i) and (ii) of Proposition 2 are equivalent. By way of contrast this is no longer the case for (ii) and (iii): the familiar five-element lattice such that 0 < a, b, b < c, a, c < 1

is a pseudocomplemented meet-semilattice such that x V x* = 1 for all x but of course it is not Boolean. This shows that the "additive structure" of a Boolean system B is essential for the Law of Excluded Middle to make B a Boolean algebra.

BOOLE'S BOOLEANNESS

9

Proposition 2 gives equational conditions which make a Boolean system a Boolean algebra; next, we add to these a number of identical implications which have the same import. Proposition 3. The following are equivalent for any Boolean system B: (i) B is a Boolean algebra. (ii) For all x £ B, I — x = 0 implies I = x. (in) For all x, y € B, x(\ — y) = 0 implies x < y. (iv) For all x,y,z £ B, xy < z implies x < z + (1 - y ) ( l — z). (v) For all x, y, z e B, z < xy and x(y — z) = 0 implies x + (y — z) = (x- z)+y. Proof, (i) => (ii). Immediate consequence of the fact that here 1 = x + (1-ar). (ii) => (i). To begin with, y = 0 whenever y(x + (1 — x ) ) = 0 in any Boolean system: the latter implies 0 = xy(x + (1 — x ) ) = xy

as well as xy + (y - xy) = 0

and hence y = 0. Now, applying this to the identity

we obtain that 1 — (x + (1 — x ) ) = 0, and by the present hypothesis this implies 1 = x + (1 — x), proving (i) by Proposition 2. (i) => (iii). Again, since y + (1 — y) — I here we have

x = xy + x(l -y) so that x = xy, that is: x < y, whenever x(l — y) = 0. (iii) =£> (i). The general identity (1 — (1— x ) ) ( l — x) =0 implies that l - ( l - z ) = (1 - (1 - x))x = x, and (i) follows by Proposition 2. (i) => (iv). In any Boolean algebra, x A y < z implies x < z V (~ y) = z V ((~ y) A (~ z))

for the complement of ~ y of y, and this is exactly (iv). (iv) =>• (i). Applying (iv) to the case x = 1, y = 0, and z arbitrary we obtain 1 = z + (1 — z) and hence (i) by Proposition 2.

10

BANASCHEWSKI

(i) =>• (v). In a Boolean algebra, the stated premiss amounts to z = x/\y, and hence ~ z = (~ x)V(~ y), so that yA(~ z) = yA(~ z) andxA(~ z) = z A ( ~ y) from which it follows that x V ( j / A ( ~ 2)) = xVy = (a;A(~ z)) \Jy. (v) =>• (i). Using the given condition for z = x and y = 1 one obtains x + (1 - x) = I so that (i) by Proposition 2. D Remark 5. Note that, at the level of prepositional logic, the above (iii) represents the principle of proof by contradiction: if p and not-q> is false then p implies q. On the other hand (iv) makes z + (1 — y)(l — z) the relative pseudocomplement y —> z, that is, the element t such that x A y < z iff x < t. Finally, (v) reads as a form of associativity if y — z is also expressed as (—z) + y, as Boole indeed does. Remark 6. Interestingly, by way of contrast, there are no identities, or identical implications, which characterize the Heyting algebras among the Boolean systems. This can be shown be exhibiting a Boolean system which is not a Heyting algebra, but which is the union of an ascending chain of Boolean subsystems that are Heyting algebras. For this, take the lattice obtained by adding the chain {^ | n = 1,2,...} to the top of the fiveelement lattice such that

0 < c< a, b < d and remove d. This provides a Boolean system B, and for each n the elements 0, a, b, c, i , . . . , ^, 1 form a Boolean subsystem such that Bn C Bn+i and B is the union of these Bn. Moreover, each Bn is a distributive lattice and hence a Heyting algebra while B itself is not even a lattice. One might add that the relation of Heyting algebras to Boolean systems in general is somewhat analogous to their relation to pseudocomplemented bounded distributive lattices. There, again, there are no identities or identical implications that single out the Heyting algebras - in obvious contrast to the Boolean algebras given by the identity x\/x* = 1 or, alternatively, x** = x. A final comment, concerning "Laws of Thought" and the Law of the Excluded Middle. In his discussion of the intuitive significance of 1 — x (III 14) Boole leaves no doubt that he accepts this law at an informal level, essentially on the basis of its accepted validity for propositions. On the other hand, in Chapter V, he states without comment that 1, expanded with respect to x, gives x + (1 — x) (V12) while later, arguing in general that the sum of the distinct constituents

BOOLE'S BOOLEANNESS

11

of any Boolean expansion is 1, he calls the initial case of the two constituents x and 1 — x "evident" (V14). Now, if this is interpreted as "evident consequence of the laws explicitly stated in Chapters II and III", our discussion shows this is incorrect: it can only be made evident by referring back to the remarks on the intuitive meaning of 1 - x, but that really amounts to the implicit acceptance of x + (1 — x) — I as a further law. Thus, the decisive feature of Boolean class logic, as understood nowadays, made its original appearance in a surprisingly surreptitious way. This is all the more remarkable in view of the pride of place accorded to the counterpart of the Law of the Excluded Middle, the Law of Contradiction rendered in the form x(l — x) = 0, and the great emphasis given to the fact that it can be derived from the basic rules stated earlier: slightly paraphrasing III 15, x = x2 = 0 + x2 implies 0 = x — x2 = x(l — x). That a calculation within his system should yield this particular result Boole considers especially significant because, as he points out, the law in question is "the most certain of all principles" and "the source of all the other axioms" (Boole refers to Aristotle, Metaphysics III, 3 but the reference actually seems to be Metaphysics F, 1005bl8). Given this emphasis, together with the subsequent discussion which referes to the Law of Contradiction as "dichotomy", it seems plausible that Boole might have considered the Law of the Excluded Middle a consequence of, if not actually synonymous with, the Law of Contradiction - which, expressed in slightly different terms would amount to a confusion between the latter and the Principle of Proof by Contradiction. It certainly seems this was a prevalent notion at one time (see Eisler[3] ), before Brouwer's critique of classical mathematics and the subsequent development of intuitionist logic shed new light on it. The exact origins of this error, which is not explicitly stated by Aristotle (Hitchcock[5]), seem well worth investigating. Acknowledgements. I am indebted to D. L. Hitchcock of the Department of Philosophy, McMaster University, for information concerning Aristotle, to J. S. Kirkaldy of the Institute for Materials Research, McMaster University, for stimulating my interest in the question treated here, to G.H. Moore for alerting me to the work of Hailperin, and to Tatjana Schonwalder of the Fachbereich Philosophie und Okonomie, LudwigMaximilians-Universitat Miinchen, for the reference to Eisler[3]. Also, financial assistance by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

12

BANASCHEWSKI

References [1] G. Boole, The mathematical analysis of logic (1847). Reprinted by Basil Blackwell, 1948. [2] G. Boole, An investigation of the laws of the laws of thought (1854). Reprinted by Dover Publications, Inc, 1951. [3] Eisler, Wortertuch der philosophischen Begriffe. Mittler, Berlin, 1907. [4] T. Hailperin, Boole's Logic and Probability. Studies in Logic and the Foundation of Mathematics, Volume 8, North-Holland, 1976. [5] D.L. Hitchcock, Oral communication, March 2003. [6] W. Kneale, Boole and the algebra of logic. Notes and Records of the Royal Society of London 1 2 (1956), 53-63. [7] E. Schroder, Vorlesungen iiber die Algebra der Logik. Teubner Leipzig 1890. [8] H.M. Sheffer, A set of five independent operations for Boolean algebras. Trans. Amer. Math. Soc. 11 (1913), 481-488.

HOMEOMORPHICALLY CLOSED NEARNESS SPACES H. L. BENTLEY Department of Mathematics University of Toledo Toledo, Ohio, 43606, USA E-mail : [email protected]

JOHN W. CARLSON Professor Emeritus Dean Emeritus University of South Dakota

In Memoriam: John W. Carlson (1940-2003) died after this paper was submitted but before it appeared. The concept of homeomorphically closed nearness spaces is introduced and used to characterize the strict TI extensions of a topological space that have the property that each homeomorphism on the underlying space lifts to a homeomorphism on the extension. Mathematics Subject Classifications (2000): 54E17, 54C20. Keywords: nearness space, extension, homeomorphism

Introduction

It is well-known how the Stone-Cech compactification can be constructed by means of the completion of a certain uniform structure. Similarly, nearness spaces (a generalization of uniform spaces) provide a setting par excellence for constructing strict TI extensions of topological spaces. Just as in the case of uniform spaces, nearness spaces have a completion and this completion is the device by means of which every strict TI extension of a TI space can be reconstructed [He74a], [BH76]. While the completion of a nearness space is pleasant in several ways [BH01], it does fail to be functorial in the non-regular case. Thus, for the extension of maps one needs to impose some extra condition in addition to uniform continuity. Several such extra conditions are presented in [BeOl]. In this paper we are concerned with the extension of a map which is a homeomorphism with respect to the underlying topological structure. We define a nearness space W to be homeomorphically closed provided 13

14

BENTLEY AND CARLSON

that every homeomorphism with respect to the underlying topology of W is uniformly continuous. In contrast to the pathology that arises from having to impose complicated extra conditions (such as was done in [BeOl]), we shall show in this paper that for a concrete homeomorphically closed TI nearness space, every homeomorphism (wrt the underlying topology) possesses a unique extension to the completion. Prom that fact, the result advertised in the abstract is an easy consequence. Nearness spaces and extensions of topological spaces This section contains only previously known results. Our purpose here is to formulate the concepts, notations, and results that we shall need for the presentation of our new results that appear in the following sections. Top denotes the category of topological spaces and continuous maps, Tops denotes the full subcategory of Top whose objects are the symmetric1 topological spaces. Nearness spaces were introduced by Herrlich in 1974 [He74a; He74b; He74c; He88]. These "topological-type" structures can be defined using either one of four concepts: uniform covers, near collections, far collections, or micromeric (or cauchy) collections. For our purposes here, the near collections are usually the most convenient. An explanation of our use of the alphabet may be in order. We usually use the letters X or Y to denote topological spaces and the letters W or V to denote nearness spaces. However, this convention is not rigid. In fact, as will be mentioned below, any symmetric topological space is also a nearness space. 1

Definitions

A nearness space is a set W endowed with a concept of certain collections of subsets of W being near, subject to the following five axioms: (Nl) If a collection A corennes2a near collection B then A is near. (N2) If r\A ^ 0 then A is near. 1

A symmetric topological space is one which satisfies x G c\{y} V between nearness spaces is said to be unifomly continuous provided whenever A is near in W, fA = { jA \ A € A } is near in V. Near denotes the category of nearness spaces and uniformly continuous maps. The basic references on nearness spaces are [He74a; He74b; He74c; He88]. Specific results relating nearness spaces to extensions of topological spaces can be found in [BH76; Be92] and surveys of the extensive literature appear in [He83; BHH98]. For some applications of nearness spaces to special classes of maps on the real numbers, see [BH78]. The closure operator defined in (N5) is a topological closure operator and the topological space so determined is called the underlying topological space of W and is denoted by TW. The underlying topological space of a nearness space is always symmetric, and every symmetric topological space has a compatible nearness structure on it defined by (T)

A is near

iff

Dcl.4^0.

16

BENTLEY AND CARLSON

Condition (T) defines a full embedding of Tops into Near, and we assume that this full embedding is an actual inclusion3 of Tops as a full subcategory of Near. Thus a symmetric topological space is a nearness space that satisfies condition (T) and between such nearness spaces, uniform continuity coincides with continuity. We call a nearness space W topological provided it satisfies condition (T). We say that a nearness space W is TI provided its underlying topological space is a TI-space. As mentioned above, a nearness space has a completion. A detailed presentation of the completion appears in [He74a]. For our purpose here, we shall need only the definition together with the characterization of the completion that appears in [Be77] (see also [BH01]). The details follow. 3

Definitions

A uniformly continuous map f : W —» V between nearness spaces is said to be 1. initial provided that whenever A is a collection of subsets of W such that fA is near in V, then A is near in W. 2. strict provided that if B is a collection of subsets of V that is far in V there exists a far collection A in W such that c\v fA corefines B. 3. dense provided fX is a dense subset of V. 4. an extension provided f is initial, dense, and is an injection. It turns out that when X and Y are topological, a map / : X —> Y is strict iff { cly fA \ A C X } is a base for the closed subsets of Y. Thus, the concept of strictness of maps between nearness spaces is a generalization of the usual notion between topological spaces. However, it may be worthwhile to point out that for nearness spaces W and V we have / : W -> V is strict

=>•

/ : TW -> TV is strict

but the converse implication does not hold. For example, let X denote IR2 with its usual topology and let W denote H2 with its usual metric uniformity. Then the identity map / : X —* ~VW is strict but / : X —> W 3

In essence, what we have is an alternative axiomatization of symmetric topological spaces by means of the structure consisting of the collections which are near, i.e., those collections that are required to satisfy the five nearness axioms plus condition (T).

NEARNESS SPACES

17

is not. To see this, note that a hyperbola together with one of its asymptotes form a two element collection that is near in W but not in X. 4

Definitions

Let W be a nearness space. 1. A nonempty collection A of subsets of W is said to be a cluster provided it is a maximal near collection. 2. W is said to be complete provided that for each cluster A on W there exists x e W such that A={AcW\x£dA}. We are now in a position to state the characterization of the completion W7* of a nearness space W. W* has as its underlying set of points the set of all clusters on W and has a nearness structure defined by: fi is near in W*



U { flu; | u G fi } is near in W

[He74a]. We have a map e : W —> W* defined by e(x) = {AcW\x£c\A}. The completion W* of a nearness space W is determined up to uniform isomorphism [Be77] by the facts that: 1. W* is a complete TI nearness space. 2. e : W —> W* is a strict, initial, and dense map. Thus, if V is a complete TI nearness space and if u : W —> V is a strict, initial, dense map then there exists a unique uniform isomorphism g : V -» W* such that W

•I \V

>

W*

a

commutes. Par abuse de language, in case W is a TI nearness space, we usually shall assume that the canonical map e : W —> W* is an inclusion. No difficulty

18

BENTLEY AND CARLSON

will arise from this convention since when W is TI the target restriction e : W —> e[W] is a uniform isomorphism. For a fuller discussion of the completion of a nearness space, see [BH01]. Our main interest here is in the investigation of strict extensions of topological spaces and the extension of maps in that setting. Nearness spaces are a perfect tool for studying strict extensions as Theorem 7 below shows. 5

Definition

A nearness space is said to be concrete provided that every near collection is a subset of some cluster. 6

Proposition [Be75]

A nearness space W is concrete iff its completion W* is topological. As a preamble to the next theorem, we observe that if W is a concrete TI nearness space, then e : TW —> W* is a strict extension of TW (recall that since W* is topological we have W* = T(W*)). The beautiful theorem of Herrlich that characterizes the strict TI extensions of a topological space is the following one: 7

Theorem [He74a; Theorem 6.3]

If X and Y are TI topological spaces and u : X —» Y is a strict extension then there exists a nearness space W with the same underlying set as X and the same underlying topology as X (i.e., TW = X) such that W is concrete and there exists a unique homeomorphism h : Y —* W* such that the triangle W

Y

h

>

W

commutes. In the above theorem, the nearness structure on W is defined by the condition A is near in W iff fl cly uA ^ 0.

NEARNESS SPACES

19

This nearness structure is said to be the one induced by the extension. Section 11 of [BH76] gives a category theoretic formulation of the relationship between strict extensions of topological spaces and completions of nearness spaces. The two categories involved are Ext and Compl which are defined as follows: Objects of Ext are strict TI extensions u : X —> Y of topological spaces and morphisms are pairs (/, g) of continuous maps such that X -^ Y

f X'

u'

>

Y'

commutes. Objects of the category Compl are pairs (W, W*) where W is a TI nearness space and W* is its completion; morphisms of Compl are uniformly continuous maps / : W* —> V* such that fW c V. The precise statement of the relationship between Ext and Compl is the following theorem. For the definition of equivalence of categories in the sense intended in the following theorem see [AHS90; 3.3]. 8

Theorem [BH76; Theorem 11.6]

The category Ext is equivalent to the category Compl under the correspondence described as follows: C : Ext —> Compl E : Compl -> Ext

where C(u : X —» Y) is the nearness space (and its completion) induced by the extension u : X —> Y and we define E(W, W*) to be the strict extension TW —> T(W*) involving the underlying topological spaces.

Extensions of homeomorphisms We are now ready to present one of our main results which is on the extension of a homeomorphism h : TW —> TV to an isomorphism h : W* —> V*.

20

9

BENTLEY AND CARLSON

Definition

If W and V are nearness spaces then we Jet Ti.(W, V) denote the group of all uniform isomorphisms h : W —> V , where the group operator is, as usual, composition of maps. In case W = V we write H(W) = H(W, W) . If W is a subspace of V we let Hw(V) = {h£ H(V) \ hW = W } . Recall that we are regarding symmetric topological spaces as topological nearness spaces, and for a map h : X —> Y between symmetric topological spaces, continuity coincides with uniform continuity. Thus in this case, H(X, Y) is the group of all homeomorphisms h : X —> Y . Hicks and McKee [HM91] have characterized those nearness spaces W for which each permutation is uniformly continuous and in that case H.(W) is the collection of all permutations of W . In the sequel, when the spaces involved are TI and when it simplifies the exposition, we shall assume that an extension u : X —> Y is an inclusion. Thus we shall say simply "Y is a TI extension of X" . 10

Proposition

Let Y be a strict TI extension of a topological space X and let W be the concrete nearness space induced by this extension. Then

Proof: This result is actually an immediate consequence of Theorem 8 above as follows: A homeomorphism h : Y —> V with hX = X delivers an isomorphism (h\X, h) of Ext and since C : Ext —> Compl

E : Compl —» Ext is an equivalence of categories, (h\X, h} induces an isomorphism h:(W,W*)-+

(W,W)

of Compl. Also, h\W = h\X e H(W). Therefore

{h\X

NEARNESS SPACES

21

To show the reverse inclusion, let h : W —> W be a uniform isomorphism. Since the completion W* of W is defined purely in terms of nearness concepts, it is evidently the case that h : W —> W "lifts" to a uniform isomorphism h* : W* —> W* . Therefore we have an isomorphism h* in the category Compl. This isomorphism corresponds to an isomorphism (h, h) : (X,Y) -» (X,Y) in the equivalent category Ext. Therefore h € HX(Y} and since h\X = h the proof is complete. d Theorem 10 shows that the uniform isomorphisms on W are precisely the homeomorphisms on X = TW that can be lifted to a homeomorphism on Y . Thus a homeomorphism on X is a uniform isomorphism on W iff it lifts to a homeomorphism on Y . Theorem 10 establishes a one-to-one correspondence between the groups H(W) and H.x(Y). The following theorem shows that these two groups are isomorphic. 11

Theorem

Let Y be a strict TI extension of X and let W be the concrete nearness space induced by this extension. Then H(W) and 7ix(Y) are isomorphic groups. Proof: This result is an immediate consequence of Theorem 8 above and the results in Section 1 1 of [BH76] . The main source of possible confusion here is one of notation. Thus, for the sake of this proof, let the extension be denoted by the more complete notation e : X —> Y. In Section 11 of [BH76], a functor F : Ext —* Top is denned by putting F(e : X —» Y) = X and > 9) — f where (/, g) is as in the following commutative diagram:

X

-!U Y

X'

-> u'

Y'

(Recall the definition of the category Ext that appears before Theorem 8.) One must exercise some care with the functor F since as is shown in [BH76], it is neither full nor faithful, but fortunately for our purposes

22

BENTLEY AND CARLSON

here, it is injective on the preimage of homeomorphisms.4 Thus we can proceed as follows: Let S : U(W) -» UX(Y) be denned by S(g) = f where in the above diagram we take X' = X, Y' — Y, and e' = e. Then S(g) = F ( f , g ) and since F is injective on homeomorphisms, S is also an injection. Proposition 10 says that S is a surjection. Therefore S is a bijection. The fact that F is a functor implies that S is a group homomorphism as follows:

=

F(f1,gl)oF(h,g2) D

In order to characterize the strict extensions X —> Y that have the property that every homeomorphism on X can be lifted to a homeomorphism on Y , the following concept is needed. 12

Definition

A nearness space W is said to be homemorphically closed provided every homeomorphism h : TW —* ~TW is a uniformly continuous map h: W -* W. 13

Theorem

Let W be a, nearness space. Then the following two statements are equivalent: 1. W is homeomorphically closed.

2. Proof: That 2 implies 1 is obvious. Assume W is homeomorphically closed. Note that in all cases we have H(W) C H(~fW). Let h £ H(TW). Then h~l £ W(TW). This means that h~l : ~TW —> 1W is a homeomorphism, and since W is homeomorphically closed, it follows that h~l : W —> W is a uniformly continuous map. Therefore h € 7i(W) . LJ 4

Perhaps it is worthwhile to point out a misprint on line 1 of page 174 of [BH76]: The map / : X -> Y should be rather the map / : X -> X'.

NEARNESS SPACES

23

The following theorem is the main result of this paper. It characterizes the strict TI extensions X —> Y such that every homeomorphism on X lifts to a homeomorphism on Y . 14

Theorem

Let Y be a strict TI extension of X and let W be the concrete nearness space induced by this extension. Then every homeomorphism on X lifts to a homeomorphism on Y iff W is homeomorphically closed, Proof:

Suppose that every homeomorphism on X lifts to a homeomorphism on Y. Let h 6 H(X) and let h £ H(Y) denote the lift of h. Then h & Hx(Y) which by virtue of Proposition 10 implies that h = h\X belongs to H(W). Hence ft(TW) = U(W) and by Theorem 13 we have that W is homeomorphically closed. Suppose now that W is homeomorphically closed. Then H(W} — and by Proposition 10,

Hence

H(X) = W(TW) = { h\X Thus every homeomorphism on X lifts to a homeomorphism on Y . 15

d

Corollary

Let W be a concrete TI nearness space. Then every homeomorphism on TW lifts to a homeomorphism on W* iff W is homeomorphically closed. We end this section by definitions and a theorem that show how our results can be applied to a continuous map that is not a necessarily a homeomorphism. 16

Definitions

Let X and X' be topological spaces and let f : X —> X' and g : X —> X' be continuous maps. 1. g is called left equivalent to f if there exists h € H(X]

such that

24

BENTLEY AND CARLSON

2. g is called right equivalent to f if there exists k £ Ti.(X') such that 9 = kf. 3. g is called equivalent to f if there exists h € 'H(X) and k 6 H(X') such that g = kfh. Clearly, each of the relations defined above is an equivalence relation on C(X, X'), the set of all continuous maps from X to X'. 17

Theorem

Let X, Y, X', Y' be topological spaces such that Y is a strict TI extension of X and Y' is a strict TI extension of X'. Let W and W respectively be the concrete nearness spaces induced by the corresponding extensions. Let f : X —» X' be a continuous map that lifts to some continuous map Y -> Y'. Let g : X -> X'. Then 1. If g is left equivalent to f and W is homeomorphically closed then g lifts to some continuous map Y —> Y'. 2. If g is right equivalent to f and W is homeomorphically closed then g lifts to some continuous map Y —> Y'. 3. If g is equivalent to f and W and W are both homeomorphically closed then g lifts to some continuous map Y —> Y'. 18

Theorem

Let X be a symmetric topological space. Let £ denote the set of all homeomorphically closed nearness structures on X. Then C with the usual order forms a complete lattice.

Examples

Let F : Tops —* Near be a concrete functor such that for each symmetric topological space X we have X = TFX and for each continuous map / we have Ff and / are the same underlying function. Then the image F[Tops] is a subcategory of Near, all of whose objects are homeomorphically closed. This observation is trivial since a homeomorphism, being continuous, is sent to a uniformly continuous map. In this section we will describe five

NEARNESS SPACES

25

different classes of nearness spaces that are homeomorphically closed and are determined by a functor in an analogous manner. As a first example, note that every symmetric topological space is homeomorphically closed since in this case continuity coincides with uniform continuity. The functor in this case is the inclusion. 19

Definition [Ca84a]

The Pervin nearness structure on a symmetric topological space X is the one that is defined by the condition: A collection A is near provided cl A has the finite intersection property. The resulting nearness space is denoted byXp. For TI -spaces X , the corresponding Pervin nearness space Xp is induced by the Wallman compactification of X . 20

Definition [Ca84b]

The Lindelof nearness structure on a symmetric topological space X is the one that is defined by the condition: A collection A is near provided cl A has the countable intersection property. The resulting nearness space is denoted by XL . XL may fail to be concrete, but in those cases where it is concrete (and when X is a TI space), its completion is a Lindelof extension that is a subspace of the Wallman compactification of X . In the next two examples, the nearness structure is most directly described using uniform covers (recall the definition in Definition 1). 21

Definition [Ca91a]

The point finite nearness structure on a symmetric topological space X is the one that is defined by the condition: A cover U of X is uniform provided it is refined by a point finite open cover of X . The resulting nearness space is denoted by may fail to be concrete, but in those cases where it is concrete (and when X is a TI space), its completion is, under suitable conditions (see [Ca91a; Theorem 3.8]), a metacompact extension that is the smallest metacompact subspace of the Wallman compactification of X that contains X .

26

BENTLEY AND CARLSON

22

Definition [Ca94]

The locally finite nearness structure on a symmetric topological space X is the one that is defined by the condition: A cover U of X is uniform provided it is refined by a locally finite open cover of X . The resulting nearness space is denoted by may fail to be concrete, but in those cases where it is concrete (and when X is a TI space), its completion is, under suitable conditions (see [Ca94; Theorem 3.12]), a paracompact extension that is the smallest paracompact subspace of the Wallman compactification of X that contains X . 23

Proposition

Let X be a symmetric topological space. homemorphically closed nearness spaces.

Then X P ,X L ,-^PF,^LF are

REFERENCES [AHS90] J. Adamek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories. John Wiley and Sons, New York (1990). [Ba64] B. Banaschewski, Extensions of topological spaces. Can. Math. Bull. 7 (1964) 1-22. [Be75] H. L. Bentley, Nearness spaces and extensions of topological spaces. In: Studies in Topology, Nick M. Stavrakas and Keith R. Allen, editors. Academic Press, New York (1975) 47-66. [Be77] H. L. Bentley, Normal nearness spaces. Quaestiones Math. 2 (1977) 23-43 and 513. [Be91] H. L. Bentley, Paracompact spaces. Topology and its Appl. 39 (1991) 283 297. [BeOl] H. L. Bentley, Extensions of maps from dense subspaces. In: Categorical Perspectives, Jiirgen Koslowski and Austin Melton, editors. Birkhauser, Boston (2001) 151-176. [BH76] H. L. Bentley and H. Herrlich, Extensions of topological spaces. In: Topology, Proceedings of the Memphis State University Conference, 1975, Stanley P. Franklin and Barbara V. Smith Thomas, editors. Marcel Dekker, New York (1976) 129-184. [BH78] H. L. Bentley and H. Herrlich, The real and the reals. General Topology and Appl. 9 (1978) 221-232. [Be92] H. L. Bentley, Strict extensions of TI spaces. In: Recent Developments in General Topology and its Applications, International

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Conference in Memory of F. Hausdorff (1868-1942), W. Gahler, H. Herrlich, and G. Preuss, editors. Akademie Verlag, Berlin (1992) 3345. [BHH98] H. L. Bentley, H. Herrlich, and M. Husek, The historical development of uniform, proximal, and nearness concepts in topology. A chapter in the book: Handbook of the History of General Topology, Volume II, C. E. Aull and R. Lowen, editors. Kluwer Academic Publishers, Dordrecht, (1998) 577-629. [BH01] H. L. Bentley and H. Herrlich, On the strict completion of a nearness space. In: Proceedings of the 8th Prague Topological Symposium 1996. Electronically published at: http//at.yorku.ca/p/p/a/a/00.htm (1999) 11-21. Also published in hard form in: Quaestiones Mathematicae 24 (2001) 39-50. [Ca84a] J. W. Carlson, Pervin nearness spaces. Topology Proc. 9 (1984) 7-30. [Ca84b] J. W. Carlson, Lindelof nearness spaces. In: Categorical Topology, Proceedings of the International Conference held at the University of Toledo, Toledo, Ohio, USA, August 1-5, 1983, H. L. Bentley, H. Herrlich, M. Rajagopalan, H. Wolff, editors. Helderman Verlag, Berlin (1984) 185-196. [Ca91] J. W. Carlson, Metacompact nearness spaces. Topology Proc. 16 (1991) 17-28. [Ca94] J. W. Carlson, Locally finite nearness spaces. Topology Proc. 19 (1994) 63-77. [He74a] H. Herrlich, A concept of nearness. Gen. Topol. Appl. 4 (1974) 191-212. [He74b] H. Herrlich, On the extendibility of continuous functions. General Topology and Appl. 4 (1974) 213-215. [He74c] H. Herrlich, Topological structures. In: Topological Structures I, Proceedings of a Symposium, organized by the Wiskundig Genoot- schap of the Netherlands on November 7, 1973, in Honour of J. de Groot 1914-1972,, Mathematisch Centrum, Amsterdam (1974) 59-122. [He83] H. Herrlich, Categorical topology 1971-1981. In: General Topology and its Relations to Modern Analysis and Algebra V, Proceedings of the Fifth Prague Topological Symposium 1981, J. Novak, editor. Heldermann Verlag, Berlin (1983) 279-383. [He88] H. Herrlich, Topologie II: Uniforme Raume. Heldermann Verlag, Berlin (1988). [HM91] T. L. Hicks and Rhonda McKee, Near-completely homgeneous spaces. Math. Japonica 36 (1991) 159-164.

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THE TENSOR PRODUCT OF ORTHOMODULAR POSETS

REINHARD BORGER FernUniversitdt Hagen Fachbereich Mathematik Liitzowstr. 125 58084 Hagen E-mail: Reinhard. [email protected] Uni-Hagen. de We show that the tensor product of orthomodular posets always exists if the tacit assumption 0 ^ 1 in the definition of orthomodular posets is dropped. This yields a symmetric monoidal category of orthomodular posets. Mathematics Subject Classifications 81P10

(2000): 06C15, 18A40, 18D10,

Keywords; Orthomodular posets, (bi-)additive map, tensor product, representable functor, symmetric monoidal category

Introduction

A major purpose of orthomodular lattices is the attempt to an algebraic treatment of the mathematical foundations of quantum dynamics. The main example is the orthomodular lattice of orthogonal projections (i.e. idempotent hermitian operators) of a (complex) Hilbert space; this lattice is isomorphic to the lattice of closed vector subspaces of the Hilbert space. But this isomorphism cannot be described purely algebraically; in general, the idempotents of a unital (not necessarily commutative) ring do not form a lattice, as opposed to the commutative case, where the idempotents always form a Boolean algebra. Since in general binary joins and meets do not seem to have a physical interpretation, it looks more appropriate to work with orthomodular posets. Usually, an orthomodular poset is defined as a poset with top element 1 and bottom element 0 and some more structure; most authors do not mention explicitly the hypothesis 0 ^ 1 . With this tacit assumption, Foulis and Bennett [1] conclude that the tensor product of orthomodular posets does not exist in general, because they give an example of an orthomodular 29

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poset X (with 0 ^ 1), for which they obtain 0 = 1 in X ® X. Later several authors tried to overcome this difficulty by relaxing the definition. Foulis and Bennett weakened the notion of orthomodular poset to the notion of orthoalgebra and showed that the tensor product always exists if there exists a unitial biadditive map between the given algebras, where they also assumed 0 ^ I in every orthoalgebra. So, if one admits 0 = 1 for orthoalgebras, the tensor product of orthoalgebras clearly always exists. We shall see below that this also holds for orthomodular posets. In comparison, removal of the condition 0 ^ 1 weakens the definition only very slightly; the only new examples are one-element orthomodular posets. It turns out that this modification yields that the tensor product always exists, and then it becomes a bifunctor, which gives rise to a symmetric monoidal structure an the category of orthomodular posets. If one considers orthomodular posets as logics (cf. [4]), one might want to exclude the case 0 ^ 1 because this means incensistency. But from the mathematical point of view it looks easier and clearer if one allows 0 = 1 because then the tensor product becomes a bifunctor, which even satisfies an associative and a commutative law. If we forbid 0 = 1 , then the tensor product is only partially defined, and the formulation of the above statements requires distinctions of several cases. So we prefer to allow 0 = 1 in the definition and ask whether 0 ^ 1 holds in concrete applications. Unfortunately, our proof uses the Adjoint Functor Theorem and does not give insight into the structure of the tensor product; in particular, it does not answer the question of when the tensor product collapses to one point. These questions remain interesting, and in general the structure of the tensor product seems to be quite complicated. On the other hand, we gain the advantage that the machinery of symmetric monoidal categories is applicable.

1. Orthomodular Posets We consider a poset with a least element 0 and an involutive antitone map X —> X, x i-> x-1 (i.e. or-1-1- = x for all x and yL < XL for x < y). Then obviously, 1 := Ox is the largest element of X. For x, y e X we define X-Ly : € {0,1}). Now assume that the join C of A0,Ai exists. Then we have A^ < C, (/j, e {0,1}) since A0,Ai are incomparable, C < Bv, C £ Bv (since #!_„ ^ £„), i.e. C < Bv (v & {0,1}). In particular, this implies 1 < rk C < 2, which is impossible because rk C must be an integer. Open Problem Let R be a unital ring with involution. Is the set of all idempotents which are stable under the involution always a lattice in the mentioned order? The answer seems to be no, but it might be difficult to find a counterexample. We shall also need the following construction: For an orthomodular poset X and for a € X, consider X\a := {x € X; x < a}. Then X\a is poset in the induced oder with least element 0 and largest element a, and X\a becomes an orthomodular poet under the new complementation x I-* x-1' := (x + a-1)-1, in particular O x ' = a and a1-' = 0. For all x, y £ X\a, x + y in X\a is as in X\\ = X; for a = 0 we see that X\o = {0} is the trivial (one-element) orthomodular poset. 2. Categorical Properties

We want to define the category OMPos of orthomodular posets. In the previous section we denned its objects; so now we shall pay attention to the morphisms. Let X, Y be orthomodular posets and let / : X —» Y be a map. If / preserves the order and the complementation (i.e. f(x-L) = f ( x ) ± for all x € X), then / preserves orthogonality (i.e. if XQ,XI e X, xo-Lxi, then /(zo)-L/(zi))- Moreover, / also preserves 0 and 1, because OJ_0 in X yields /(0)_L/(0) in Y and thus /(O) = 0 by (OMP1) and therefore /(I) =/(O- 1 -) = QJ-= 1. We call an arbitrary map / : X —> Y additive if /(z 0 + xi) = f ( x 0 ) + f ( x i ) holds for all x0, xi € X with x0Lxi, and we call / unital if /(I) = 1. Easy examples show that preservation of order and complementation in general does not imply additivity. Proposition 2.1. For ortho-normal posets X, Y and a map f : X —> Y the following assertions hold: (i) f is a homomorphisms of orthonormal posets if and only if f is additive and unital. (ii) If f is additive, then f yields a homomorphism f : X —> ^|/(i) by codomain restriction.

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Proof, (i) The "only if" part is clear, and for the "if" part it suffices to show that / preserves oder and complementation. For XQ,XI e X with XQ < xi, (OMP3) for X yields an x? 6 X with x0±X2, XQ + x2 = xi, hence f(xQ) < f ( x 0 ) + f ( x 2 ) = f(x0 + z 2 ) = f ( x i ) . For every x e X we have x + or1 = 1 , hence f ( x ) + /(a;-1) = f(x + or1) = /(I) = 1, hence /(or1) = /(a:)1- by (v) and (vi) of 1.1. (ii): For every x 6 X we have x + x-1 = 1 in X, hence /(x) < /(x) + /(a;-1) = /(x + or1) = /(I) in r, i.e. f ( x ) & Y\fW. Thus / induces a map / : X —> y|/(j), which is unital by definition and additive by additivity of / and therefore a homomorphism by (i). D For an orthomodular poset X and for o € X, the inclusion map X\a X is always additive, but unital only in the trivial case a = 1; so it is a homomorphism only in this case. Now we can define the category OMPos, whose objects are orthomodular posets and whose morphism are homomorphisms of orthomodular posets. Clearly, the one-element orthomodular poset O := {0} is a terminal object, since for ever X the unique (constant) map X —> O is a homomorphism. On the other hand, if / : O —> X is a homomorphism, then 0 = 1 in O implies 0 = /(O) = /(I) = 1 in X and thus 0 < x < 1 = 0 and hence x = 0 for all x e X, i.e. X = O. Thus all homomorphisms with domain O are isomorphisms. The category OMPos also has an initial object, namely D := {0,1} with 0 < 1, Q-1- = 1, I-1 = 0, for every orthomodular poset X the map 0 —» 0, 1 i—> 1 is the unique homomorphisms from D to X. The forgetful functor is representable the representing object is S ':— {Q, 1, s, sx} with 0 < s < 1, 0 < s1 < 1, Ox = 1, l x = 0, s11 = s, where s and sx are incomparable. For an arbitrary orthomodular poset X and for x G H, there is a unique homomorphism 5 —> X which maps s to x, namely the map 0 H-> 0, 1 H-> 1, s H-> x, sx H-> a;1. It is easy to see that up to isomorphism (9, -D, 5 are the three smallest orthomodular poset, i.e. every orthomodular poset with at most four element is isomorphic to O, D, or S. Theorem 2.1. The category OMPos is complete, and the forgetful functor OMPos preserves all existing (possibly large) limits. Proof. Preservation of limits follows from representability. For the existence, it suffices to construct products and equalizers. For a set I and a family (Xi)i€i of orthomodular posets, the product Hie/ -^» *s *^e set-theoretic

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product with componentwise order and complementation. If /, g : X —> Y are homomorphisms of orthomodular posets, the equalizer of / and g is the set-theoretic embedding Z X, where Z := {x e X; f ( x ) = g(x)} with order and complementation inherited from X. Then Z obviously satisfies (OMP1), and (OMP2) follows immediately from additivity of / and g; for (OMP3) use (vii) of 1.1 and observe (x0 + x^)1- € Z for XQ, Xi € Z C X, XQ < Xi.

D

Observe that the product of the empty family is the terminal object O. For the sake of completeness, we like to mention that OMPos is also cocomplete, which will not be used further in this paper. The coproduct of a non-empty family (Xi)i€i of orthomodular posets can be constructed as follows. If 0 ^ 1 holds in all Xi, then first take the set-theoretic union of all Xi with order and complementation on each summand (such that elements of distinct summands are incomparable) and divide out the equivalence relation ~, where for x e Xi, y £ Xj, we have x ~ y if either x — 0 (in X^, y = 0 (in Xj) or x = 1, y = 1 or i = j, x = y. Then there is a unique orthomodular structure on the set-theoretic quotient such that the projection is a homomorphism of orthomodular posets. Then this quotient is the coproduct of (^¥j)j 6 /. If Xi = O holds for some i 6 /, it is easy to see that O is a coproduct of (-Xi)ig/. The coproduct of the empty family is the initial object D. The existence of coequalizers can be shown by Adjoint Functor Theorem techniques, but this does not yield much insight into their structure. 3. The Tensor Product For orthomodular posets X, Y, Z, we call a map b : X x Y —> Z biadditive if it is additive in each component, i.e. for all x e X the map Y —> Z, y i-> b(x,y) is additive and for all y e Y the map X —>• Z, x >-> b(x,y) is also additive. Observe that for each x G X we have b(x, 0)±b(x, 0), because OJ_0 in Y; this implies b(x, 0) = 0 by (OMP1) in Z; similarly we have 6(0, y) = 0 for all y € Y. Moreover, we call the above b unital if 6(1,1) = 1 holds. Note that this means unitality as a map from the categorical product X x Y to Z. On the other hand, biadditivity is quite different from additivity on X x Y. Lemma 3.1. For orthomodular posets X, Y, Z and for a map b : X x Y —> Z the following statements are equivalent: (i): b is biadditive and unital, and for every orthonormal poset W and

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every biadditive unital map c : X x Y —> W there exists a unique homomorphism f : Z —> W of orthomodular posets with fb = c. (ii): b is biadditive, and for every orthonormal poset W and every biadditive map c : X x Y —* W there exists a unique additive map f : Z -* W with fb = c. Proof, (i) => (ii): If c : X x Y —> W is biadditive, then c is isotone in both components, hence c(x,y) < c(l, 1) holds for all re € AT, y e Y. Thus we obtain an map c : X x Y —> Wj c (i i i) by codomain restriction, and c is clearly biadditive and also unital by definition. Now (i) yields a unique homomorphism / : X x Y —» W^i^) with / o b = c. Then it is easy to see that the composite Z —> VF| c (i,i) t-> W is the unique additive map / : Z -> W with fb = c. (ii) => (i): At first we show that b is unital. By codomain restriction we obtain an biadditive map b : X x Y —> Z\b(i,i), and the universal property (i) renders a unique additive map I : Z —> £|&(i,i) with Ib = b. But the inclusion map u : Z|b(i,i) "—> Z is also additive with b — ub = ulb, and the uniqueness part of the universal property of (i) gives that ul is the identity. In particular, u is surjective, hence 1 € Z = Z|b(i,i), i.e. 6(1,1) = 1. Now let W be an orthonormal poset and consider an arbitrary biadditive unital map c : X x Y —> W. Then from (i) we obtain a unique additive map / : Z —> W with fb = c. But then / is unital because /(I) = /6(1,1) = c(l,l) = 1; thus / is a homomorphism by 2.1. Moreover, / is the only homomorphism with f b = c because it is even the only addive map satisfying this equation. D Proposition 3.1. Let X be an orthonormal poset. (i) For the initial orthonormal poset D = {0,1} the map bo '• X x D —-> X with bfj(x, 0) := 0 and &o(x,l) := x for all x e X has the universal properties of 3.1. (ii) For the terminal orthonormal poset O = {0}, the unique (constant) map X x O —> O has the universal properties of 3.1. Proof, (i) It is easy to see that 60 is biadditive and unital. For an orthonormal poset W and a unital biadditive map c, there is at most one homomorphism / : X —» W with fb = c, because each such / must satisfy f ( x ) = fb(x, 1) = c(x, 1) for all x € X. On the other hand, the map / : X —> W, x i—> c(x, 1) is obviously additive and unital and therefore a

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homomorphism, which satisfies fb — c because f b ( x , l ) = f ( x ) = c(x, 1) and f b ( x , 0) = 0 = c(x, 0) for all x € X. (ii) The (constant) map X x O —» O is trivially biadditive. For an orthomodular poset W and a biadditive map c : X x O —> W, the equation 0 = 1 in O implies 0 = c(l, 0) = c(l, 1) = 1 in W and hence W =* O, which immediately proves our claim. D Theorem 3.1. For any orthomodular posets X, Y there exist an orthomodular poset Z and a unital biadditive map b : X x Y —» Z such that for each orthomodular poset W and each unital biadditive map c : X x Y —> W , there is a unique homomorphism f : Z —> W with fb = c. Proof. We use the Adjoint Functor Theorem (cf. [2], [3]). Consider the functor F : OMPos —> Set that assigns to each orthomodular poset W the set of all unital biadditive maps from X x Y to W. Then a routine check shows that F preserves products and equalizers and therefore all small limits. In order to verify the solution set condition, fix an infinite cardinal a such that both X an Y and thus also X x Y are of cardinality < a. Then for each orthomodular poset W and each unital biadditive map c : X x Y —> W the set-theoretic image AQ of c is of cardinality < a. If An C W is defined and of cardinality < a, then Bn := AnU{x-L \ x € An C W} is of cardinality < a + a = a, and An+i ••= {xi + . . . + xr | r 6 Nu{0}, xi,...,xr &Bn, Xi^Xj

for i ^ j}

is of cardinality < £ r€Nu{0} ar = 1 + £ reN a < a • N0 = a. Thus W0 := UneNu{o} ^" = UneN u {o} Bn C W is a subset of cardinality ^ X)neNu{0} a =a-

Moreover, Wo is closed under complementation and orthogonal sums; thus Wo is an orthomodular poset in the structure inherited from W, and the inclusion Wo ^-> W is a homomorphisms. On the other hand, by codomain restriction, c induces a unital biadditive map X x Y —> WQ. This shows that every unital biadditive bilinear map X x Y —> W factors over a unital biadditive map X x Y —> WQ via a homomorphism WQ —> W, such that the cardinality of Wo is < a. But there are only a set of cardinals /3 < a, and for each such /3 and each set WQ of cardinality f3 there are only a set of structures on WQ as an orthomodular poset and of maps X x Y —> WQ. Thus up to isomorphism in the comma category there is only a set of unital biadditive maps X x Y —> WQ with WQ of cardinality < a. If 7 is the

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cardinality of this set, the for each set I of cardinality there is a solution set of cardinality < 7*. In particular, F satisfies the solution set condition. Hence F has a left adjoint and is therefore representable. If we choose Z to be the representing object and b : X x Y —> Z to be the element of FZ corresponding to the identity of Z under the natural isomorphism, we see that Z and b have the desired properties. D Observe that the above proof does not give much insight into the concrete structure. In particular, it is not clear whether there exists an n e N such that An = An+i always holds in the proof of the solution set condition. This should give some indications about how complicated it is to describe the orthonormal poset generated by some given subset. By the universal property, in Theorem 3.1 Z and b are unique up to a canonical isomorphism; so we can introduce some notations. For X, Y as in 3.1, we call some chosen Z with the properties there the tensor product of X and y, we write it as X®Y, and we denote the map b : X x Y —> Z = X®Y by (x, y) H-> x y. It follows from standard arguments that ® behaves like a functor in each component, und thus we obtain a functor OMPos x OMPos -» OMPos, (X, Y)^X®Y. From 3.1 we obtain canonical isomorphisms X ® D = X and X O = O for all orthomodular posets X. Lemma 3.2. (i) For all orthomodular posets X, Y there is a unique isomorphism X Y —> Y X with x y >-> y x for all x e X, 7 / 6 Y. (ii) For all orthonormal posets X, Y, Z there is a unique isomorphism (X Y) ® Z -> X ® (Y ® Z) with (x ® y} ® z H-» x ® (y ® z) for all

x ex, y e Y, z e z. Proof, (i) Since the map X x Y —> Y y, (x, y) H-> y ® x is biadditive and unital, the universal property of X ® Y renders a unique homomorphism s : X®Y -» y ®X with s(x ® y) = y <S> x for all x e X, ye Y. Since this argument is symmetric, there is also unique homomorphism t : Y ® X y — > X ® y are the identity maps, i.e. s and t are isomorphisms with s"1 = t.

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(ii) : For fixed z e Z, the map XxY -> X(g)(y®Z), (x,y) i-> z(yz) is biadditive; thus there exists a unique additive map hz : X <S> Y —> X ® (y Z) with /i z (x ® y) = a; (y 2) for all x e X, y e y. For z 0 , zi e Z with zo-Lzi it follows that y®zo-Lyzi and a;(yzo)-La;(yzi) for all a; € X, y e y, and for all x e X, y 6 y we obtain hZQ(x®y) + hZl(x z ® (y ® z0) + z (y Zi) = X (y z0 + y zx) = z ® (y (z0 + hzo+zi(x®y). Themap/i 2o +/i Zl : X®Y -» X(yZ), u i-> hzo(u)+hzi(u) is clearly additive; thus the map (X(g>y) x Z —> X(y®Z), (u, z) i-> ftz(u) is biadditive, and the universal property renders a unique additive map s : (X®Y)®Z -> X(yig>Z) with s(u®z) = /i z (u) for all u G Jfy, z e Z. In particular, we have

s((x ® y) (X y) Z with t(x ® (y ® z)) = (x ® y) ® z for all x e X, y 6 Y, z £ Z. Going through the construction, the uniqueness part of the universal properties shows that st : X ® (y Z) -» X ® (y ® Z) and is : (X ® Y) ® Z -> ( X y) (g) Z

are identity homomorphisms, therefore s is an isomorphism and t = s 1 .D Theorem 3.2. OMPos is a symmetric monoidal category with tensor product anrf unit object D = {0,1} in the canonical way. Proof. For orthomodular posets X, Y, Z let the associativity map (X ® y) Z —> X (y ® Z) be the unique homomorphism with (x y] ® z i—» x ® (y ® z) for all x 6 X, y € y, z e Z. Obviously, this isomorphism is natural in all arguments. Similarly, the symmetry X ®Y —> Y d§ X is the unique map with x y >—> y ® x for all x 6 X, y e y. Moreover, the isomorphisms X £) —» X are given b y x < E > l | - ^ a ; , x ® 0 i—> 0 for all x e X; its inverse is given by x \—> a; (gi 1 for all x e X. Similarly we have an isomorphism D ® X = X. All these maps are natural in all arguments. The proof of the coherence conditions is technical but straightforward. D

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It is quite easy to see that OMPos is not closed in the structure; the functor OMPos —* OMPos, X H-> X Y has a right adjoint only if Y = D. Indeed, it this functor has a right adjoint, then it preserves the initial object D, i.e. Y = D ® Y = D. References [1] Foulis, D. and Bennett, M.K, Tensor products of orthoalgebras, Order 10 (1993), 271-282. [2] Herrlich, H. and Strecker, G.E., Category theory, Allyn and Bacon, Boston 1973. [3] Mac Lane, S., Categories for the working mathematician, Graduate Texts in Mathematics, Springer, Berlin 1972.. [4] Ptak, P. and Pulmanova, S., Orthomodular Structures as Quantum Logics, Fundamental Theories of Physics 44, Kluwer 1991.

ON A WEAK FORM OF THE BLUMBERG PROPERTY

WING-SUM CHEUNG Department of Mathematics University of Hong Kong Pokfulam Road, Hong Kong email: [email protected] YU-TING LIN AND SHIOJENN TSENG* Department of Mathematics Tamkang University Tamsui, Taipei, Taiwan, 25137, R.O.C. email: [email protected] GEORGE E. STRECKER Department of Mathematics Kansas State University Manhattan, KS 66506 USA email: streckerQmath. ksu. edu

In this paper we modify and extend the techniques used by Blumberg in his classic paper of 1922. In particular, we formulate a weakening of the Blumberg property and show that a wide class of not necessarily metrizable spaces has this property. Mathematics Subject Classifications 54C30, 54C50, 54E52

(2000): 26A15, 26A21, 54C08,

Keywords: real-valued function, (weak) Blumberg property, (weak) Blumberg space, separable T\ Baire space, dense approach, exhaustible approach

"This work is partially financially supported by the National Science Council of the Republic of China under the projects NSC87-2115-M032-004 and NSC88-2115-M032010 41

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1. Introduction In 1922 Blumberg [1] showed that every separable completely metrizable space X has the following surprising property: B: for each real-valued function f : X —» R there is a dense subset D of X such that the restriction f \D is continuous. Since then B has been called the Blumberg property and spaces having it have been called Blumberg spaces. The theory of Blumberg spaces is well-developed. For example, in 1960 Bradford and Goffman [2] showed that every Blumberg space is a Baire space and that the converse is true if the Baire space is metrizable. In 1977 Weiss [5] showed that there are compact Hausdorff spaces that are not Blumberg. In this paper we consider spaces that need not be metrizable and develop a theory for them analogous to that developed in [1]. We define weak Blumberg spaces, show that every separable T\ Baire space is weak Blumberg, and provide examples of non-metrizable weak Blumberg spaces. 2. The Weak Blumberg Property Throughout let (X, r) be a topological space (often denoted by X alone), let R be the real numbers with the usual topology, let Q be the rational numbers and N be the natural numbers. For each p e X, the set of open neighborhoods of p will be denoted by A/"p. Definition 2.1. Let 9^ be a binary relation between open sets and elements, i.e., 91 C T x X. For U a member of T and p an element of X, Uytp means that the open set U has the relation SH to the element p, i.e., (U,p) e IH. We say that the relation 9f is closed if for every subset A of X and every U G T, t/fHs for all s e A imply U9lp for all p e A. Definition 2.2. A partial neighborhood of p, often denoted by N R and eac/i pair of real numbers r\ and r% with ri < TI the relation 9^ir2 C. T x X given by: p if and only ifpeU and there exists some q £ U such that ri < is a closed relation. Proof. Suppose that A C X and U e T such that E7Sft£jT.2s for all s & A. Then for each such s we have s e t / and there is some q 6 (7 such that ri ^ /(