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Post-modern algebra. (English) Zbl 0946.00001

This book, as its somewhat jocular title is meant to suggest, is intended by its authors to be a successor to the famous textbook of van der Waerden. It includes many topics now important in applications but either unknown, or considered unimportant, in van der Waerden’s days. These include automata, categories, loops, quasigroups, semigroups. The chapters are: 0. Introduction; I. Groups and Quasigroups; II. Linear Algebra; III. Categories and Lattices; IV. Universal Algebra. This list gives a good idea of the coverage, which, as one might expect from two well-known experts in the field, is comprehensive.
I think that giving a lecture course based on this book would be an interesting and rewarding experience. The exercises are numerous and well-thought out, though access to some solutions would be helpful; several times I found myself wondering exactly what the authors had in mind. Perhaps they could consider adding some hints and solutions to the web-site mentioned below. Also some of the examples of applications are very interesting. However, I feel that the authors missed a golden opportunity by not mentioning the importance of Galois fields in coding theory.
But I fear that students, even with the solid grounding in undergraduate mathematics that the authors say is required, would have some difficulty in working through this book unaided. I have to admit that I would have found the definition of wreath product given here totally incomprehensible, had I not already had a good acquaintance with the topic! The lack of an index of notation would also make this a very difficult book to dip into; although some notation is given in the general index; not all is, and this can make things awkward. For example, \(\widehat A\) is used to denote the diagonal of \(A\times A\); but \(\widehat V=\underline{\underline{\text{Set}}}^{V^{\text{op}}}\). Only the latter is listed in the index, and then, unfortunately, with the wrong page number.
I was saddened by the so-called “proof” of mathematical induction. I feel the authors should have been honest about it, and perhaps presented an intuitive argument, rather than write down such a circular argument.
There are not too many misprints; two are corrected in an errata slip, and others may be found on the web-site www.math.iastate.edu/jdhsmith/math/books.htm. The only one I thought might cause confusion was the omission of the word unital in the discussion of maximal ideals of a ring, which leaves the unfortunate impression that all rings have maximal ideals.
I found this an interesting book to read, and feel that anyone who works through it and works all the exercises, will obtain an excellent grounding in algebraic concepts, particularly those which have practical applications.

MSC:

00A05 Mathematics in general
08-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general algebraic systems
18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
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