Rédei / Sneddon / Stark The Theory of Finitely Generated Commutative Semigroups

1. Auflage 2014
ISBN: 978-1-4831-5594-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 368 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

The Theory of Finitely Generated Commutative Semigroups describes a theory of finitely generated commutative semigroups which is founded essentially on a single 'fundamental theorem' and exhibits resemblance in many respects to the algebraic theory of numbers. The theory primarily involves the investigation of the F-congruences (F is the the free semimodule of the rank n, where n is a given natural number). As applications, several important special cases are given. This volume is comprised of five chapters and begins with preliminaries on finitely generated commutative semigroups before turning to a discussion of the problem of determining all the F-congruences as the fundamental problem of the proposed theory. The next chapter lays down the foundations of the theory by defining the kernel functions and the fundamental theorem. The elementary properties of the kernel functions are then considered, along with the ideal theory of free semimodules of finite rank. The final chapter deals with the isomorphism problem of the theory, which is solved by reducing it to the determination of the equivalent kernel functions. This book should be of interest to mathematicians as well as students of pure and applied mathematics.

Autoren/Hrsg.

Rédei, L.

Sneddon, I. N.

Stark, M.

Gravett, K. A. H.

Inhaltsverzeichnis

1;Front Cover;1
2;The Theory of Finitely Generated Commutative Semigroups;4
3;Copyright page;5
4;Table of Contents;6
5;Preface;8
6;Introduction;10
7;Chapter I. Kernel functions and fundamental theorem;16
7.1;1. Preliminaries;16
7.2;2. Axioms I—V of the kernel functions;25
7.3;3. Fundamental theorem;27
7.4;4. Second form of Axiom V;30
7.5;5. Proof of the fundamental theorem;33
8;Chapter II. Elementary properties of the kernel functions;38
8.1;6. Set stars and ideal stars;38
8.2;7. Third form of Axiom V;42
8.3;8. Fourth form of Axiom V;46
8.4;9. The star property of the kernel functions;49
8.5;10. i'irst theorem of reciprocity;52
8.6;11. Transitivity classes;59
8.7;12. Reduction of Axiom V;72
9;Chapter III. Ideal theory of free semimodules of finite rank;77
9.1;13. Dickson's theorem;77
9.2;14. The ideals of and F°;78
9.3;15. Translation classes of ideals;86
9.4;16. Ideal lattice and principal ideal lattice;88
9.5;17. Direct decompositions in F and F°;92
9.6;18. The height of ideals of F;99
9.7;19. The maximal condition in the ideal lattice of 7^;101
9.8;20. Semiondomorphisms of the ideal lattices of F°;103
9.9;21. Certain congruences in commutative cancellative semigroups;108
9.10;22. jP"-congruences by ideals;110
9.11;23. Second theorem of reciprocity;117
9.12;24. The classes for an ideal of F;129
9.13;25. The set of classes by an ideal of F;142
10;Chapter IV. Further properties of the kernel functions;146
10.1;26. The kernel of -congruences or kernel functions;146
10.2;27. Translated kernel functions;155
10.3;28. Finiteness of the range of values of the kernel functions;159
10.4;29. Classification of the kernel functions;163
10.5;30. The kernel functions of first degree;164
10.6;31. The enveloping kernel function of first degree;186
10.7;32. The kernel functions of first order;188
10.8;33. Finite definability of finitely generated commutativesemigroups;194
10.9;34. The lattice of kernel functions;203
10.10;35. Connection of an F-congruence with the values of thekernel function belonging to it;208
10.11;36. The submodules of F°;211
10.12;37. Finite commutative semigroups;212
10.13;38. Numerical semimodules;216
10.14;39. Investigation of the kernel functions **in the little;221
10.15;40. The numerical semimodules attached to the kernelfunctions;222
10.16;41. The kernel functions of first rank;236
10.17;42. The maximum condition in the lattice of kernel functions;242
10.18;43. The normals of a kernel function;247
10.19;44. Splitting kernel functions;252
10.20;45. The kernel functions of second order;258
10.21;46. The kernel functions of second dimension;297
10.22;47. Degenerate kernel functions;338
11;Chapter V. Equivalent kernel functions;342
11.1;48. Preparations for the solution of the isomorphism problem;342
11.2;49. Submodules of equivalent relative to F;349
11.3;50. Equivalent kernel functions;356
12;Appendix;360
12.1;51. The case of semigroups without a unity element;360
13;Index;364
14;Other titles in the series;366