The Concise Handbook of Algebra

ISBN-13: 9789401732697 / Angielski / Miękka / 2013 / 618 str.

Alexander V. Mikhalev; G. F. Pilz

Opis

It is by no means clear what comprises the "heart" or "core" of algebra, the part of algebra which every algebraist should know. Hence we feel that a book on "our heart" might be useful. We have tried to catch this heart in a collection of about 150 short sections, written by leading algebraists in these areas. These sections are organized in 9 chapters A, B, ..., I. Of course, the selection is partly based on personal preferences, and we ask you for your understanding if some selections do not meet your taste (for unknown reasons, we only had problems in the chapter "Groups" to get enough articles in time). We hope that this book sets up a standard of what all algebraists are supposed to know in "their" chapters; interested people from other areas should be able to get a quick idea about the area. So the target group consists of anyone interested in algebra, from graduate students to established researchers, including those who want to obtain a quick overview or a better understanding of our selected topics. The prerequisites are something like the contents of standard textbooks on higher algebra. This book should also enable the reader to read the "big" Handbook (Hazewinkel 1999-) and other handbooks. In case of multiple authors, the authors are listed alphabetically; so their order has nothing to do with the amounts of their contributions.

Spis treści

Preface. A Semigroups. A.1. Ideals and Green's Relations; L.N. Shevrin. A.2. Bands of Semigroups; L.N. Shevrin. A.3. Free Semigroups; L.N. Shevrin. A.4. Presentations and Word Problems; P.M. Higgings, N. Ruskuc. A.5. Simple Semigroups; L.N. Shevrin. A.6. Epigroups; L.N. Shevrin. A.7. Periodic Semigroups; L.N. Shevrin. A.8. Finite Semigroups and Pseudovarieties; J. Almeida. A.9. Regular Semigroups; P.G. Trotter. A.10. Completely Regular Semigroups; J.M. Howie. A.11. Inverse Semigroups; P.G. Trotter. A.12. Separated Transformation Semigroups; K.D. Magill. A.13. Matrix Semigroups; J. Okninski. A.14. Subsemigroup Lattices; A.J. Ovsyannikóv, L.N. Shevrin. A.15. Varieties of Semigroups; L.N. Shevrin, M.V. Volkov. A.16. Compact Semigroups; K.H. Hofmann. A.17. Applications of Semigroups; G.F. Pilz, et al. B Groups. B.1. Abelian Groups; A.V. Mikhalev. B.2. p-Groups: Basics and the Coclass Project; C.R. Leedham-Green. B.3. p-Groups: Other Approaches; C.R. Leedham-Green. B.4. Soluble Groups; H. Lausch. B.5. Permutation Groups; P.J. Cameron. B.6. Lie Groups; J. Hilgert. B.7. Lie Groups and Differential Equations; P.J. Olver. B.8. Simple Groups; R. Solomin. B.9. Free and Relatively Free Groups; G.Baumslag. B.10. Free Products, Amalgamated Products, and HNN Extensions; G. Baumslag. B.11. Word Problems in Groups; D. Holt. B.12. Combinatorial Group Theory; G. Baumslag. B.13. The Burnside Problems; Rostislav, et al. B.14. Automatic and Hyperbolic Groups; D. Holt. B.15. Actions of Finite Groups; P. Paule. B.16. Wreath Products; J.D.P. Meldrum. B.17. Automorphism Groups of Algebraic Systems; B.I. Plotkin. B.18. Frobenius Groups; P. Fleischmann. B.19. Covered and Fibered Groups; C.J. Marson. B.20. Subgroup Lattices of Finite Abelian Groups; F. Vogt. B.21. Varieties of Groups; A.A. Iskander. B.22. Groups and Absolute Geometry; H. Karzel. B.23. Groups for Particle Physics; G. Kalmbrach. C Rings, Modules, Algebras. C.1. Commutative Algebra I: Ideal Decompositions; R. Gilmer. C.2. Commutative Algebra II: The Krull Dimension; R.Gilmer. C.3. Factorization and Decomposition of Polynomials; J. von zur Gathen. C.4. Element Factorization in Integral Domains; D.F. Anderson. C.5. Dedekind and Prüfer Domains; M. Fontana, I.J. Papick. C.6. Local Rings; T.Y. Lam. C.7. Semilocal Rings; T.Y. Lam. C.8. Cohen-Macaulay Rings; J. Herzog. C.9. Rings of Formal Power Series; A.A. Tuganbaev. C.10. Automorphisms of Polynomial Algebras and the Jacobian Conjecture; J.-T. Yu. C.11. R