Books by Norman R. Reilly

Introduction to Applied Algebraic Systems
Norman R. Reilly (2010)

This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. The book offers a brief introduction to elementary number theory as well as a fairly complete discussion of major algebraic systems (such as rings, fields, and groups) with a view of their use in bar coding, public key cryptosystems, error-correcting codes, counting techniques, and elliptic key cryptography. This is the only entry level text for algebraic systems that includes an extensive introduction to elliptic curves, a topic that has leaped to prominence due to its importance in the solution of Fermat's Last Theorem and its incorporation into the rapidly expanding applications of elliptic curve cryptography in smart cards. Computer science students will appreciate the strong emphasis on the theory of polynomials, algebraic geometry and Groebner bases. The combination of a rigorous introduction to abstract algebra with a thorough coverage of its applications makes this book truly unique

Exercise Solutions Available

A complete set of solutions to approximately 850 exercises in An Introduction to Applied Algebraic Systems is available to instructors who adopt the book.

If you are interested, send an e-mail to Dr. Reilly: nreilly at math.sfu.ca. Please include your full name, title, department, institution, street address, and course name. You have our promise that the information will be kept strictly confidential.

Completely Regular Semigroups
Mario Petrich; Norman R. Reilly (1999)

In the great range of special classes of semigroups, regular semigroups take a central position from the point of view of richness of their structural "regularity". The principal special classes of regular semigroups are inverse semigroups and completely regular semigroups with a great diversity of their various generalizations. These statements are corroborated amply by the semigroup literature and are reflected somewhat by the books on semigroups. Among the latter there are books on the general theory of semigroups, in all of which regular semigroups figure prominently, and one book that specializes exclusively in inverse semigroups.

Completely regular semigroups come also under the label of unions of groups because they are unions of their (maximal) subgroups. This property which makes them relatively close to groups reflects itself in any attempt at describing their structure. At the other extreme, they resemble bands (idempotent semigroups) which gives them some combinatorial features. More specialized bands like semilattices and rectangular bands play a fundamental role in any structural description of bands and thus also of completely regular semigroups. These special classes of completely regular semigroups, namely groups, bands, and semilattices, together with completely simple semigroups, indeed play a basic role in the general structure theory of completely regular semigroups.

This book is devoted exclusively to the theory of completely regular semigroups: structure, congruences, and varieties. This ought to cover all the principal aspects of the relatively large material now available concerning completely regular semigroups.