Inverse semigroups : the theory of partial symmetries

INVERSE SEMIGROUPS

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INVERSE SEMIGROUPS The Theory of Partial Symmetries Copyright © 1998 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 981-02-3316-7

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This book is dedicated to the memory of my father Alfred Mark Lawson (1930-1993) 'We are all things that make and pass, striving upon a hidden mis­ sion, out to the open sea.' H. G. Wells

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Preface Introduction An appreciation of symmetry appears to be a feature of the human mind: even the earliest human artifacts were shaped in ways which transcended the purely utilitarian. The mathematical theory of symmetry grew out of investigations into the solutions of polynomial equations. Developing ideas of Lagrange and Abel, Galois discovered that the nature of these solutions could be determined by studying certain symmetries of the equation. It was from this work that the concept of a group as a mathematical device for measuring symmetry gradually evolved. The theory of groups is one of the most successful branches of algebra with applications which range from error-correction in communication systems to the existence of elementary particles. But although group theory is certainly concerned with symmetry, it is by no means the case that the converse is true. This was highlighted in spectacular fashion in 1984, when Shechtman et al announced the discovery of a metallic phase whose diffraction pattern showed icosahedral symmetry, something which had long been assumed impossible on the basis of a result from group theory known as the 'crystallographic restric­ tion'. The fault lay not in group theory, but in the assumptions underlying the translation of the intuitive idea of symmetry into a mathematical one. As in all translations, something was lost in the process. In this book, we concentrate on just one aspect of our intuitive idea of symmetry which fails to be captured by groups: namely, the relationship be­ tween the parts and the whole. To see that group theory does not capture this relationship in a satisfactory way, one need only consider those natural forms, modelled by fractals, which are not highly symmetrical in the classical sense, but do possess self-similarity properties in which the whole is repeated at smaller scales. We instinctively feel that such forms possess symmetry but, due to their global irregularity, not of a kind which can be detected by groups. Self-similarities are examples of what we term partial symmetries, by which we mean symmetries between the parts of a structure. Partial symmetries are vii

VU1

Preface

common in mathematics not only on their own terms, but also in situations where the principal interest may be centred on symmetries; this is because the first step in constructing a symmetry is frequently to construct a partial symmetry and then to extend it to a symmetry. Galois' theory of equations makes essential use of this idea. The mathematical structures which encode information about partial sym­ metries are certain generalisations of groups called inverse semigroups. Like groups, inverse semigroups first arose in questions concerned with the solutions of equations, but this time in Lie's attempt to find the analogue of Galois the­ ory for differential equations. The symmetries of such equations form what are now termed Lie pseudogroups, and inverse semigroups are, several times removed, the corresponding abstract structures. In addition to their early ap­ pearance in differential geometry, inverse semigroups have found a number of other applications in recent years including: C*-algebras; tilings, quasicrystals and solid-state physics; combinatorial group theory; model theory; and linear logic. One of my aims in writing this book was to show that inverse semigroups had the potential to act as a unifying concept for applications of partial sym­ metries, in much the same way that groups have acted as a unifying concept for applications of symmetries. Structure of the book The book is notionally divided into three parts: (I) The first two chapters present the two main themes of the book: the relationship between inverse semigroups and partial symmetries; and the relationship between partial symmetries and global symmetries. For the cognoscenti, Chapter 1 revolves around the Wagner-Preston representa­ tion theorem, and Chapter 2 around the McAlister covering theorem. (II) Chapters 3, 5, 6, 7 and 9 present the essentials of inverse semigroup theory, concentrating particularly on the theory of ^-unitary and 0-JBunitary inverse semigroups. Chapter 9 is rich in applications of inverse semigroups, most notably to the theory of tilings and quasi-crystals. (Ill) Chapters 4, 8 and 10 contain detailed applications of category theory to inverse semigroups. Chapters 4 and 8 incorporate Ehresmann's theory of ordered groupoids into mainstream inverse semigroup theory by applying it to the theory of ^-unitary covers and idempotent pure extensions. Chapter 10 is, in some ways, a continuation of Chapter 1, in that it shows in what sense every inverse semigroup can be regarded as a semigroup of partial symmetries of some structure. This work has its origins in

Preface

IX

Girard's use of inverse semigroup theory in his geometry of interaction programme for linear logic. In order to read this book, I have assumed that the reader has followed a basic undergraduate mathematics course. A familiarity with category theory, sufficient to understand what is meant by an equivalence of categories, would be useful at a number of points in the text, but only essential for the theory developed in Chapters 4, 8 and 10. Acknowledgements This book would not have been completed without the goodwill and assistance of many friends and colleagues. Although I had been toying with the idea of writing a book for some time, the impetus to start work came from Douglas Munn. Douglas's own contributions to inverse semigroup theory, together with those of Don McAlister, form the backbone of this subject as a glance at the bibliography will show. I have been lucky in obtaining support for my research from a number of institutions. I began my study of Ehresmann's Oeuvres when I was a Junior Research Fellow at Lincoln College, Oxford. My thanks to the Fellows of Lin­ coln, in particular Vivian Green and David Edwards, for electing me to the Fellowship, and to Peter Neumann and Hilary Priestley who supported my first painful efforts to formalise what I knew about semigroup theory. Subse­ quently, I spent a year in Darmstadt, Germany supported by the Royal Society in Prof. Karl Hofmann's research group where I was able to continue the work initiated in Oxford interspersed with Kaffee und Kuchen. The British Council supported a visit to Australia, and the Hungarian Academy of Sciences and the Royal Society supported a visit to Hungary. In Australia, I discussed with David Easdown, Tom Hall, Gordon Preston, Bob Sullivan, and Peter Trot­ ter many aspects of inverse and general semigroup theory. In Hungary, early drafts of the first few chapters were tried out by the Budapest-Szeged Semi­ group Seminar: namely, Pham Ngoc Anh, Laszlo Marki, Gyorgy Pollak and Maria Szendrei. Special thanks are due to Laci and Maria, whose comments caused me to rethink my approach, and whose corrections saved me from many embarrassing errors. The contents of Chapter 9 owe a special debt to Johannes Kellendonk, who patiently explained his work to me in both Bangor and Berlin; I am grateful to Prof. Dr. U. Pinkall, Sprecher of Sonderforschungsbereich 288, at the Technische Universitat, Berlin for the invitation to visit Berlin and for the financial support provided by the DFG. Paul Goldstein first alerted me to Johannes's work during a Gregynog Pure Mathematics Colloquium here at the University of Wales.

X

Preface

Des FitzGerald read a number of chapters and his comments helped to sharpen up the presentation considerably. Conversations with John Fountain, Vicky Gould, Peter Jones and Don McAlister helped clarify my thinking about the main arguments of this book. Thanks are also due to my research students: Peter Hines and Helen James; Peter first alerted me to the work of Girard and subsequently took the first steps in trying to understand self-similarity from an inverse semigroup-theoretic perspective, and Helen's eagle-eye spotted many typos. John Hickey helped out with numerous references and background information. Many others supported this project by providing offprints of their work or by making encouraging noises from the side-lines. A number of colleagues at Yr Ysgol Fathemateg, Prifysgol Cymru, Bangor assisted with the production of this book: Julie Thistlethwaite and Sian Jones typed some of the chapters, and Garreth Roberts and Chris Wensley provided computer expertise; diolch yn fawr iawn i chi. Lakshmi Narayan of World Scientific waited patiently for this book to see the light of day. Finally, I would like to express my indebtedness to two outstanding books: Howie's 'Introduction to semigroups' is the canonical text for anyone inter­ ested in semigroup theory, whereas Petrich's 'Inverse semigroups' has been an invaluable reference throughout the preparation of this book. The cover design was supplied by Michael Roesgen and Johannes Kellendonk and shows part of an aperiodic tiling derived by projection from Z 7 .

Contents Preface

vii

1

1 1 10 15 17 36 39

Introduction to inverse semigroups 1.1 The origins of inverse semigroups 1.2 Pseudogroups and local structures 1.3 Symmetry 1.4 Abstract inverse semigroups 1.5 Representation theorems 1.6 Notes on Chapter 1

2 Extending partial symmetries 2.1 Partial symmetries 2.2 iT-unitary covers 2.3 Congruences 2.4 The minimum group congruence 2.5 Notes on Chapter 2 3 The 3.1 3.2 3.3 3.4 3.5 4

natural partial order The associated groupoid Green's relations Primitive inverse semigroups The bicyclic monoid Notes on Chapter 3

47 48 56 58 62 69 75 76 82 93 97 103

Ordered groupoids 4.1 Inductive groupoids 4.2 Ordered groupoids from *-semigroups 4.3 Ordered congruences 4.4 Notes on Chapter 4 XI

107 108 116 124 130

xii

Contents

5

Extensions of inverse semigroups 5.1 The Kernel-trace description of a congruence 5.2 Idempotent-separating extensions 5.3 A-semidirect products 5.4 Inverse w-semigroups 5.5 Notes on Chapter 5

133 134 138 147 160 167

6

Free inverse semigroups 6.1 The existence of free inverse semigroups 6.2 Solving the word problem 6.3 The structure of free inverse semigroups 6.4 Munn normal form 6.5 Notes on Chapter 6

171 171 174 185 187 194

7

E-unitary inverse semigroups 7.1 Division theorems 7.2 The P-theorem 7.3 The Mobius inverse monoid 7.4 F-inverse semigroups 7.5 F-inverse covers 7.6 Notes on Chapter 7

197 198 208 216 221 226 230

8 Enlargements 8.1 A coordinate-free P-theorem 8.2 F-unitary covers 8.3 The maximum enlargement theorem 8.4 Applications 8.5 Notes on Chapter 8

233 233 239 248 259 270

9 O-F-unitary inverse semigroups273 9.1 The structure of a class of O-F-unitary inverse semigroups . . . 274 9.2 Kellendonk's topological groupoid 278 9.3 Polycyclic monoids 285 9.4 McAlister semigroups 304 9.5 Tiling semigroups 318 9.6 Notes on Chapter 9 327 10 Category actions and inverse semigroups 10.1 Inverse semigroups from category actions 10.2 A coordinatisation theorem 10.3 Category actions from inverse semigroups 10.4 Functors between systems and semigroups 10.5 Equivalent systems

331 332 335 341 345 347

Contents 10.6 Composing the functors 10.7 Special cases 10.8 Notes on Chapter 10

xiii 353 358 369

Bibliography

373

Index

405

Chapter 1

Introduction to inverse semigroups In this chapter, we trace the development of inverse semigroup theory from its origins in the theory of pseudogroups of transformations to the WagnerPreston representation theorem. We also examine the role of inverse semi­ groups in mathematics, arguing that they capture aspects of our intuitive understanding of symmetry in a more complete way than groups.

1.1

The origins of inverse semigroups

Inverse semigroups were first explicitly defined by V. V. Wagner in a short note published in 1952 [419], and independently by Gordon Preston in 1954 [325], [326] and [327]. Earlier work by Clifford [45], [46] and Rees [335], [336] can be viewed as contributions to inverse semigroup theory, as can the contempora­ neous paper of Clifford [47]. Much of Charles Ehresmann's work in the 1940s and 1950s was also concerned with inverse semigroup theory, although written from a different mathematical perspective. Inverse semigroups were introduced as part of the legacy of two nineteenthcentury mathematical enterprises: Klein's Erlanger Programm and Lie's theory of infinite continuous groups. The Erlanger Programm In the course of the nineteenth century, understanding of the nature of geom­ etry underwent a profound seachange. Euclidean geometry, which had previ­ ously occupied a privileged position, came to be seen as just one amongst a 1

2

Introduction to inverse semigroups

great variety of different geometries. But with this diversity came the problem of how these geometries should be classified and how the relationships between them should be established. The central idea of Klein's Erlanger Programm [88] was that this could be achieved using the group of symmetries of the geometry. The symmetry group of a geometry consists of all its structure-preserving bijections; thus from every geometry we can construct a group. However, Klein argued that this group could in turn be used to reconstruct the geometry. This is because the symmetry group induces an equivalence relation between geometric figures: two figures are equivalent if one can be mapped to the other by means of a symmetry. Geometric properties are properties of equivalence classes of figures rather than individuals, and so the geometry can be viewed as the invariant theory of the group. This group-theoretic classification of geometries can also be used to estab­ lish relationships between geometries. Let Q be a geometry with symmetry group G, and suppose that G' is a subgroup of G. Figures equivalent with respect to G' are equivalent with respect to G, but not conversely. Thus the subgroup G' can be used to construct a geometry Q' which makes finer distinc­ tions between figures. We say that the geometry Q' is richer than the geometry Q. In this way, the classical non-Euclidean geometries and the circle, sphere and line geometries of Mobius, Lie and Pliicker can all be constructed from projective geometry, as can the Euclidean, equiform and affine geometries. It is not surprising that Cayley was moved to write that 'projective geometry is all geometry' [21] (see also [436]); but, in fact, there were geometries which simply could not be accommodated within the group-theoretic framework. The fol­ lowing passage from the important 1932 book by Veblen and Whitehead [418] indicates one source of the difficulty: But long before the Erlanger Programm had been formulated there were geometries in existence which did not properly fall within its categories, namely the Riemannian geometries. [...] Riemannian spaces [... ] are metric spaces [... ] whose groups of automorphisms reduce to the identity. Such a geometry obviously cannot be char­ acterised by a group. J. H. C. Whitehead [433] in his obituary of Elie Cartan makes the point that, between the formulation of the Erlanger Programm and the formulation of General Relativity, the foundations of geometry were dominated by Klein's approach, but that After the discovery of general relativity, which was based on Rie­ mannian geometry, it was realized that the Erlanger Program (sic) was no longer adequate as a general description of geometry.

The origins of inverse semigroups

3

Nevertheless, the fact that the Erlanger-Programm did not work for all geometries simply spurred mathematicians into trying to generalise it. This point was explicitly made by Veblen and Whitehead: There is, therefore, a strong tendency among contemporary geome­ ters to seek a generalization of the Erlanger Programm which can replace it as a definition of geometry by means of the group concept. In other words, if the group of symmetries is not sufficient to classify the geometry perhaps there is a more general structure which can. One particular way of generalising the Erlanger Programm was based on the concept of a pseudogroup introduced by Sophus Lie. Pseudogroups In the 1880s, Sophus Lie began to study what he termed continuous groups, partly as an attempt to develop a Galois theory for differential equations [299]. In modern terminology, the objects Lie studied consisted of families of partial diffeomorphisms of a manifold defined by a system of partial differential equa­ tions which were closed under composition and inverses1. Lie distinguished two classes of such groups: the finite and the infinite. The finite continuous groups are essentially representations by diffeomorphisms of an abstract, finitedimensional Lie group. The infinite continuous groups are not groups at all but merely group-like. These structures have come to be called Lie pseudogroups (see [2], [3], [324], and [349]). A Lie pseudogroup without its differential equations is just a collection of partial homeomorphisms between the open subsets of a topological space which is closed under composition and inverses. Veblen and Whitehead [418] recognised in such structures, which they called simply pseudogroups, suitable algebraic vehicles for generalising the Erlanger Programm to the developing theory of differential manifolds: The geometric objects and the spaces which they determine are classified by means of the pseudo-group of regular point transfor­ mations, two objects belonging to the same class if, and only if, they are equivalent. It is this pseudo-group, rather than the group [...], which is relevant, because a geometric object is not necessarily de­ fined over the whole of a simple manifold [...]. This classification of geometric objects is in the spirit of the Erlanger Programm, equiv­ alence under the pseudo-group being an inevitable generalization of equivalence in the narrower sense. ' A diffeomorphism is a homeomorphism which is infinitely differentiable and whose in­ verse is also infinitely differentiable.

4

Introduction to inverse semigroups

Thus the Erlanger Programm is generalised by replacing groups of symme­ tries by pseudogroups. Such pseudogroups can be used to define many of the basic structures of differential geometry; we explain precisely how this is done in Section 1.2. Focussing on pseudogroups in this way raises a new question. Groups of transformations are representations of abstract groups, so it is nat­ ural to ask what the corresponding abstract structures are for pseudogroups. It is precisely this question which served as the impetus for denning inverse semigroups. Before describing how this question was answered, we formalise some of the terms we have been using. Partial bijections and their properties Let X and Y be any two sets. A partial function f from X to Y is a function from a subset of X to a subset of Y. The subset of X consisting of all those elements x 6 X for which f(x) is defined is called the domain (of definition) of f, which we denote by d o m / . The image of / is the subset i m / = / ( d o m / ) of Y. Two special classes of partial functions are particularly important. For any two sets X and Y there is a unique empty partial function from X to Y which we denote by Oyx- We use the term empty function to refer to any partial function of this type and allow the context to determine which one we mean. For every subset A of X the identity function on A, denoted 1A, is a partial function from X to itself. Such partial functions are termed partial identities. The partial identity function on d o m / is denoted by d ( / ) and the partial identity function on i m / is denoted by r ( / ) . The identity function 1^ on X and the identity function 1© on the empty subset of X, which is just the empty function from X to itself, are especially significant. When the underlying set X is clear, we denote these functions by 1 and 0 respectively. Let g be a partial function from X to Y and / a partial function from Y to Z. Then their composite is a partial function fog from X to Z, where the domain of / o g is given by dom(/ og) = g'1 (dom

fnimg)

and if x G dom(/ o g) then (/ o g)(x) = f{g{x)). The image of / o g is / ( d o m / n img). The case where d o m / and img have empty intersection causes no problems: / o g is just the empty function. We usually write fg rather than fog. We shall concentrate on those partial functions which induce bijections between their domains and images; we call such partial functions partial bijec­ tions. All partial identities and empty functions are partial bijections. If / is a partial bijection from X to Y then we denote by / _ 1 the partial bijection from Y to X which is the inverse of / . Thus the domain of / - 1 is i m / and

5

The origins of inverse semigroups

its image is d o m / . The composition of partial bijections is again a partial bijection. Some important properties of partial bijections are described below. Proposition 1 Let X, Y and Z be sets, and let f: X -4 Y be a partial bijection. (1) f~l f = ldom/, o, partial identity on X, and f f~l = l i m / , o, partial iden­ tity on Y. (2) For a partial bijection g: Y —> X, the equations f = fgf and g = gfg hold if, and only if, g — / _ 1 .

0) cr 1 )-^/. (4) l ^ l s = IAHB = I B I A for all partial identities \A and \B where A,B C X.

(5) (gf)~x

= f~lg~l

for any partial bijection

g-.Y^Z.

Proof (1) Observe that x £ d o m ( / - 1 / ) precisely when x 6 d o m / . Thus the domains of / and / _ 1 / are the same. But f~lf is the identity function on its domain and so / _ 1 / = ldom/; as required. The other result is proved similarly. (2) Suppose that / = fgf and g = gfg. Let y € dom g and put x — g(y). Then x = {gfg)(y), so that x = g{f(x)). But g is a partial bijection, and so y = f(x). Hence x = /_1(2/)> which gives g C f~l. Now let y 6 d o m ( / _ 1 ) and put x = f~l{y), so that f(x) = y. Then (fgf){x) - y so that f(g(y)) = y. But / is a partial bijection and so g(y) = x, which gives / _ 1 C g. Consequently, we have shown that f~l — g. The converse is clear. (3) This follows from (2). (4) We show that I A I B = IAOB- Let x G d o m ^ l s ) . Then (l,ilB)(a;) = l ^ l ^ a : ) ) = x. Thus l ^ l s is a partial identity. Since dom(iAls) = AnB, it follows that IA^B = 1/inB(5) We have that

gfif-'g-^gf = g(fr1)(g-19)f = rtrtX/r1)/ = gf since by (4) partial identities commute. We may similarly show that

(f-1g-1)gf(f-lg-1) Thus (gf)'1

= T V

1

by (2).

=

f-1g-1■

The collection of all partial bijections between sets forms a category, but we shall mainly be interested in the set of all partial bijections from a set X

6

Introduction to inverse semigroups

to itself. This forms a monoid2, denoted I{X), called the symmetric inverse monoid on X. Recall that an idempotent in a semigroup is any element equal to its square. An important property of symmetric inverse monoids is the following; its sig­ nificance will become apparent in Theorem 3. Proposition 2 Let I(X) be the symmetric inverse monoid on the set X. Then the idempotents of I(X) are precisely the partial identities on X. In particular, the idempotents form a commutative, idempotent subsemigroup. Proof By Proposition 1(4), every element of the form 1A is an idempotent of I(X). Suppose now that / is an idempotent. Then / = / / / , so that / _ 1 = / by Proposition 1(2). Thus / = / / = f~lf = l d 0 m/ by Proposition 1(1). ■ We can now return to the problem we posed earlier: what are the abstract structures which are represented by pseudogroups? This received two, distinct solutions. Inverse semigroups The first solution was provided independently by Wagner and Preston with their introduction of the class of inverse semigroups; these were defined to be semigroups S satisfying the following two conditions: (1) 5 is regular. This means that for every element a € S there is an element b, called an inverse of a, satisfying a = aba and b — bob. (2) The idempotents of S commute. From Propositions 1 and 2 symmetric inverse monoids really are inverse semi­ groups. Wagner and Preston then proved that every inverse semigroup could be embedded in some symmetric inverse monoid. This result is proved in Section 1.5. Soon after inverse semigroups were introduced, the following theorem was proved. Theorem 3 Let S be a regular semigroup. Then the idempotents of S com­ mute if, and only if, every element of S has a unique inverse. Proof Let 5 be a regular semigroup in which the idempotents commute and let u and v be inverses of x. Then u = uxu = u(xvx)u =

(ux)(vx)u,

2 It may be worth pointing out at this point that a semigroup is just a set with an associative binary operation, whereas a monoid is a semigroup with an identity.

The origins of inverse semigroups

7

where both ux and vx are idempotents. Thus, since idempotents commute, we have that u = {vx){ux)u = vxu = (vxv)xu — v(xv)(xu). Again, xv and xu are idempotents and so = v(xux)v = vxv = v.

u = v(xu)(xv)

Hence u = v. To prove the converse, we begin by proving that in a regular semigroup the product of two idempotents e and / has an idempotent inverse. Let x = {ef)' be any inverse of ef. Consider the element fxe. It is idempotent because (fxe)2

= f{xefx)e

= fxe,

and it is an inverse of ef because {fxe)ef(fxe)

= {fxe)2 = fxe and ef{fxe)ef

= {ef)x{ef)

= ef.

Now let 5 be a semigroup in which every element has a unique inverse. We shall show that ef = fe for any idempotents e and / . By the result above, f{ef)'e is an idempotent inverse of ef. Thus {ef)' = f{ef)'e by uniqueness of inverses, and so {ef)' is an idempotent. Every idempotent is self-inverse, but on the other hand, the inverse of {ef)' is ef. Thus ef = {ef)' by uniqueness of inverses. Hence ef is an idempotent. We have shown that the set of idempo­ tents is closed under multiplication. It follows that fe is also an idempotent. But ef{fe)ef = {ef){ef) = ef, and fe{ef)fe = fe since ef and fe are idem­ potents. Thus fe and ef are inverses of ef. Hence ef = fe. ■ It follows that inverse semigroups are precisely the semigroups 5 in which for each element s G S there exists a unique element s" 1 E S such that s = ss~ls and s _ 1 = s~1ss~1. The element s _ 1 is called the inverse of s in S. An inverse subsemigroup of an inverse semigroup is a subsemigroup closed under inverses. A Pseudogroup is an inverse semigroup of partial homeomorphisms between open subsets of a topological space 3 . Ordered groupoids We have already indicated that inverse semigroups were only one solution to the problem of abstractly characterising pseudogroups. The second solution, due to Charles Ehresmann, took as its starting point a different definition of the composition of partial bijections. 3

There is little consensus in the literature on the definition of pseudogroups; this is about the most generous.

8

Introduction to inverse semigroups

Let g be a partial function from X to Y and / a partial function from Y to Z. The restricted product of / and g, denoted by / • g, is defined only when d o m / = img, in which case / • g = fg. In terms of our notation, the restricted product / ■ g is defined precisely when d ( / ) = r(g). We shall call the product we defined earlier the full product when we wish to distinguish it from the restricted product. The two ways of composing partial bijections stem from two different ways of viewing partial bijections. When they are considered with respect to the restricted product, we are actually regarding them as total functions on their domains of definition, whereas the full product is the product of partial bijec­ tions regarded as partial functions. Partial bijections under the restricted product do not form a semigroup, instead they form a groupoidm the sense of category theory; that is, a category in which every morphism is invertible. The theories of groupoids and inverse semigroups are not equivalent. However, in addition to the restricted product of partial functions, a partial order can be defined on the collection of partial functions. If / and g are two partial functions from X to Y such that dom / C dom g and f(x) = g(x) for all x £ dom / then we write / C g and say that / is a restriction of g. Some important properties of this partial ordering in terms of the full product are described below. Proposition 4 Let f,g:X—>Ybe Y. (1) JffCg

partial bijections between the sets X and

thenf-'Cg-1.

(2) Let h,k: Y -> Z be partial bijections between the sets Y and Z. If f C g and h C k then hf C kg. (3) / C j precisely when there is a partial identity lA f = giA-

£ I{X)

such that

Proof The proofs of (1) and (2) are straightforward. (3) Suppose that f Cg. Let lA = f~l f where A = d o m / by Proposition 1(1). By (2) above, / C g\A. In particular, d o m / C d o m ^ l ^ ) . Let x £ d o m ^ l ^ ) . Then 1.4(0;) is defined so that x € d o m / . Thus d o m / = d o m ^ l ^ ) . Hence

f = gUConversely, suppose that / = g\A for some partial identity 1^. Let x E dom(/). Then f(x) is defined and so (glA){x) is defined. It follows that x € dom#. Thus d o m / C domg. But observe that with the above x, we

The origins of inverse semigroups

9

have that (g\A){x) = g(x). Thus / C g.

■

Whereas Wagner and Preston found axioms satisfied by the semigroup (I(X), o), Ehresmann found axioms satisfied by the structure (I(X), ■, C) with­ in the theory of ordered groupoids. These two approaches are in fact equivalent; this is proved properly in Chapter 4, but we can easily provide some insight into the nature of this equivalence here. If / is a partial function from X to Y and X' is a subset of dom / then we may define a new partial function (/ | X'), the restriction of f to X', to be the partial function from X to Y with dom(/ | X') = X' such that (/1 X')(x) = f(x) for all x 6 X'. Likewise, if Y' is a subset of im / then we define ( V | / ) , the corestriction of f to Y', to be the partial function from X to Y with d o m ( r | / ) = f~l{Y') such that ( V | f){x) = f(x) for all x 6 dom(y' | / ) . The essence of the relationship between inverse semigroups and ordered groupoids is contained in the following result. P r o p o s i t i o n s Let f, g € I{X), and put A = d o m / D img. fg =

Then

(f\A)-(A\g).

Proof Prom the definition of the restricted product dom((f\A)-(A\g))

=

dom((A\g)),

and from the definition of the corestriction dom{(A\g))

= s _ 1 (A) = g'^domf

n imp).

Thus dom((f\A)-(A\g))

=

dom(fg).

The result is now clear.

■

Concluding remarks In this section, we have tried to explain how inverse semigroups arose from an attempt to generalise the Erlanger Programm to differential geometry. How­ ever, it is important to stress that pseudogroups are just one of the ways in which the group concept is generalised in differential geometry. Kirill Macken­ zie in the introduction to his book [209] provides a broader perspective for the ideas of this section: The modern, rigorous concept of a group is far too restrictive for the range of geometrical applications envisaged in the work of Lie.

10

Introduction to inverse semigroups There have thus arisen the concepts of Lie pseudogroup, of differentiable and of Lie groupoid, and of principal bundle—as well as var­ ious related infinitesimal concepts such as Lie equation, graded Lie algebra and Lie algebroid—by which mathematics seeks to acquire a precise and rigorous language in which to study the symmetry phenomena associated with geometrical transformations which are only locally defined.

1.2

Pseudogroups and local structures

In this section, we describe how pseudogroups can be used to construct geo­ metric structures in differential geometry. In the 1950s, Ehresmann analysed this construction in terms of category theory. This work and its applications to inverse semigroup theory are described in Section 8.3. Charts and atlases At its simplest, differential geometry concerns spaces which look locally like pieces of Mn and pseudogroups provide the glue to hold these pieces together. We now introduce the mathematics which will enable us to make this idea precise. Let X and Y be topological spaces. In what follows, the space X will be the model space and our aim will be to construct geometric structures on Y which are locally like pieces of X. A chart from X to Y is a homeomorphism 4>: U -¥ V between open subsets of X and Y respectively. Thus a chart is nothing other than a special kind of partial bijection. An atlas from X to Y is a collection of charts from X to Y such that the union of the images of the charts is Y\ thus l y = \J(j>eA 0_1. If this latter condition does not hold we shall say that A is a partial atlas. If A is a partial atlas from X to Y, and B is a partial atlas from Y to Z, then BA is the partial atlas from X to Z defined by BA = {ip<j>: i> 6 B and

y. Uj — ► Vj be any charts in the (partial) atlas A from XtoY. Then we can form the partial homeomorphism

07V.: ^(Vin

V}) -► 4>jl{Vi nVj)

between the open subsets 0~1(Vj n Vj) and faj1 (VJ PI Vj) of X. Partial homeomorphisms such as these are called transition functions of the atlas. Thus the transition functions of a (partial) atlas A from X to Y belong to the pseudogroup T(X) of all partial homeomorphisms between the open subsets of X. This property can be succinctly expressed by the equation A-1 A C T(X). In order to construct geometric structures, we shall need to know under what circumstances partial bijections can be glued together. Proposition 1 Let f,g: X —> Y be partial bijections between the sets X and Y. (1) / U g is a partial function precisely when fg~l (2) fUgisa

partial bijection precisely when f~lg

is an idempotent. and fg~l

are idempotents.

(3) Let {fa: i £ 1} be a family of partial bijections from X to Y. Then [j(f>i is a partial bijection if, and only if, 4>i U ~l is a partial bijection from Y to X and

(5) Let {fa: i £ 1} be a family of partial bijections from X toY such that |J fa is a partial bijection. Then for any partial bijection fa W -> X the union \Jfa(f> is a partial bijection from W to Y, such that

((J*»)*=U**Similarly, for any partial bijection tp:Y -> Z the union |J tpfa is a partial bijection from X to Z such that

12

Introduction to inverse semigroups

Proof (1) Suppose that / U g is a partial function. Then / and g agree on d o m / D doing. By Proposition 1.1.1, the domain of the idempotent e = f~1fg~1g is just d o m / D domg. It is easy to check that fe = ge. But fe = fg-^g and ge = gf'1/. Thus fg'lg = gf~lf. Composing on the right by g~l we obtain fg~x — g{f~l})g~l, which is an idempotent. x Conversely, suppose that jg~ is an idempotent. Then fg~l = {fg~l)~l = l gf~ : and so ( / s - 1 ) ^ / - 1 ) = j / ~ x . Hence fg^gf'1 = gf~l. Multiplying by / on the right, we obtain f(g_1g) =