Algebras, Lattices, Varieties (The Wadsworth & Brooks/Cole Mathematics Series)

Volume 1 is a leisurely paced introduction to general algebra and lattice theory. Besides the fundamental concepts and elementary results, it contains severa1 harder (but basic) results that will be required in later volumes and a final chapter on the beautiful topic of unique factorization. This volume is essentially self-contained. We sometimes omit proofs, but-except in rare cases-only those we believe the reader can easily supply with the lemmas and other materials that are readily at hand. It is explicitly stated when a proof has been omitted for other reasons, such as being outside the scope of the book. We believe that this volume can be used in severa1 ways as the text for a course. The first three chapters introduce basic concepts, giving numerous examples. They can serve as the text for a one-semester undergraduate course in abstract algebra for honors students. (The instructor will probably wish to supplement the text by supplying more detail on groups and rings than we have done.) A talented graduate student of mathematics with no prior exposure to our subject should find these chapters easy reading. Stiff resistance will be encountered only in $2.4-the proof of the Direct Join Decomposition Theorem for modular lattices of finite height-a tightly reasoned argument occupying severa1 pages. In Chapter 4, the exposition becomes more abstract and the pace somewhat faster. Al1 the basic results of the general theory of algebras are proved in this chapter. (There is one exception: The Homomorphism Theorem can be found in Chapter 1.) An important nonelementary result, the decomposition of a complemented modular algebraic lattice into a product of projective geometries, is proved in $4.8. Chapter 4 can stand by itself as the basis for a one-semester graduate course. (Nevertheless, we would advise spending severa1 weeks in the earlier chapters at the beginning of the course.) The reader who has mastered Chapters 1-4 can confidently go on to Volume 2 without further preliminaries, since the mastery of Chapter 5 is not a requirement for the later material. Chapter 5 deals with the possible uniqueness of the factorization of an algebra into a direct product of directly indecomposable algebras. As examples, integers, finite groups, and finite lattices admit a unique factorization. The Jordan normal form of a matrix results from the unique decomposition of the representa- l tion module of the matrix. This chapter contains many deep and beautiful results. Our favorite is Bjarni Jónsson's theorem giving the unique factorization of finite algebras having a modular congruence lattice and a one-element subalgebra (Theorem 5.4). Since this chapter is essentially self-contained, relying only on the Direct Join Decomposition Theorem in Chapter 2, a one-semester graduate course could be based upon it. We believe that it would be possible to get through the whole volume in a year's course at the graduate level, although none of us has yet had the opportunity to try this experiment.