Symmetry is one of the most important organising principles in the natural sciences. The mathematical theory of symmetry has long been associated with group theory, but it is a basic premise of this book that there are aspects of symmetry which are more faithfully represented by a generalization of groups called inverse semigroups. The theory of inverse semigroups is described from its origins in the foundations of differential geometry through to its most recent applications in combinatorial group theory, and the theory tilings.
In recent years, it has become increasingly clear that there are important connections relating three concepts -- groupoids, inverse semigroups, and operator algebras. There has been a great deal of progress in this area over the last two decades, and this book gives a careful, up-to-date and reasonably extensive account of the subject matter. After an introductory first chapter, the second chapter presents a self-contained account of inverse semigroups, locally compact and r-discrete groupoids, and Lie groupoids. The section on Lie groupoids in chapter 2 contains a detailed discussion of groupoids particularly important in noncommutative geometry, including the holonomy groupoids of a foliated manifold and the tangent groupoid of a manifold. The representation theories of locally compact and r-discrete groupoids are developed in the third chapter, and it is shown that the C*-algebras of r-discrete groupoids are the covariance C*-algebras for inverse semigroup actions on locally compact Hausdorff spaces. A final chapter associates a universal r-discrete groupoid with any inverse semigroup. Six subsequent appendices treat topics related to those covered in the text. The book should appeal to a wide variety of professional mathematicians and graduate students in fields such as operator algebras, analysis on groupoids, semigroup theory, and noncommutative geometry. It will also be of interest to mathematicians interested in tilings and theoretical physicists whose focus is modeling quasicrystals with tilings. An effort has been made to make the book lucid and 'user friendly"; thus it should be accessible to any reader with a basic background in measure theory and functional analysis.
"this volume represents an outstanding contribution to the field. The resolute graduate student or mature researcher, alike, can find a wealth of directions for future work".Mathematical Reviews
With over 60 figures, tables, and diagrams, this text is both an intuitive introduction to and a rigorous study of finite symmetric inverse semigroups. The book presents much of the material on the theory of finite symmetric inverse semigroups, unifying the classical finite symmetric group theory with its semigroup analogue. A comment section at the end of each chapter provides new historical perspective. New proofs, new theorems and the use of multiple figures, tables, and diagrams to present complex ideas make this book current and highly readable.
Most papers published in this volume are based on lectures presented at the Chico Conference on Semigroups held on the Chico campus of the Cal ifornia State University on April 10-12, 1986. The conference was spon sored by the California State University, Chico in cooperation with the Engineering Computer Sciences Department of the Pacific Gas and Electric Company. The program included seven 50-minute addresses and seventeen 30-minute lectures. Speakers were invited by the organizing committee consisting of S. M. Goberstein and P. M. Higgins. The purpose of the conference was to bring together some of the leading researchers in the area of semigroup theory for a discussion of major recent developments in the field. The algebraic theory of semigroups is growing so rapidly and new important results are being produced at such a rate that the need for another meeting was well justified. It was hoped that the conference would help to disseminate new results more rapidly among those working in semi groups and related areas and that the exchange of ideas would stimulate research in the subject even further. These hopes were realized beyond all expectations.
This festschrift volume in honour of Donald B McAlister on the occasion of his 65th birthday presents papers from leading researchers in semigroups and formal languages. The contributors cover a number of areas of current interest: from pseudovarieties and regular languages to ordered groupoids and one-relator groups, and from semigroup algebras to presentations of monoids and transformation semigroups. The papers are accessible to graduate students as well as researchers seeking new directions for future work.
This festschrift volume in honour of Donald B McAlister on the occasion of his 65th birthday presents papers from leading researchers in semigroups and formal languages. The contributors cover a number of areas of current interest: from pseudovarieties and regular languages to ordered groupoids and one-relator groups, and from semigroup algebras to presentations of monoids and transformation semigroups. The papers are accessible to graduate students as well as researchers seeking new directions for future work.
This festschrift volume in honour of Donald B McAlister on the occasion of his 65th birthday presents papers from leading researchers in semigroups and formal languages. The contributors cover a number of areas of current interest: from pseudovarieties and regular languages to ordered groupoids and one-relator groups, and from semigroup algebras to presentations of monoids and transformation semigroups. The papers are accessible to graduate students as well as researchers seeking new directions for future work.
