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The Theory of Finitely Generated Commutative Semigroups
1st Edition - January 1, 1965
Author: L. Rédei
Editors: I. N. Sneddon, M. Stark, K. A. H. Gravett
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 5 5 9 4 - 4
The Theory of Finitely Generated Commutative Semigroups describes a theory of finitely generated commutative semigroups which is founded essentially on a single "fundamental… Read more
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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.
Preface
Introduction
Chapter I. Kernel Functions and Fundamental Theorem
1. Preliminaries
2. Axioms I—V of the Kernel Functions
3. Fundamental Theorem
4. Second Form of Axiom V
5. Proof of the Fundamental Theorem
Chapter II. Elementary Properties of the Kernel Functions
6. Set Stars and Ideal Stars
7. Third Form of Axiom V
8. Fourth Form of Axiom V
9. The Star Property of the Kernel Functions
10. First Theorem of Reciprocity
11. Transitivity Classes
12. Reduction of Axiom V
Chapter III. Ideal Theory of Free Semimodules of Finite Rank
13. Dickson's Theorem
14. The Ideals of F and F°
15. Translation Classes of Ideals
16. Ideal Lattice and Principal Ideal Lattice
17. Direct Decompositions in F and F°
18. The Height of Ideals of F
19. The Maximal Condition in the Ideal Lattice of F
20. Semiendomorphisms of the Ideal Lattices of F°
21. Certain Congruences in Commutative Cancellative Semigroups
22. F-Congruences by Ideals
23. Second Theorem of Reciprocity
24. The Classes for an Ideal of F
25. The Set of Classes by an Ideal of F
Chapter IV. Further Properties of the Kernel Functions
26. The Kernel of F-Congruences or Kernel Functions
27. Translated Kernel Functions
28. Finiteness of the Range of Values of the Kernel Functions
29. Classification of the Kernel Functions
30. The Kernel Functions of First Degree
31. The Enveloping Kernel Function of First Degree
32. The Kernel Functions of First Order
33. Finite Definability of Finitely Generated Commutative Semigroups
34. The Lattice of Kernel Functions
35. Connection of an F-Congruence with the Values of the Kernel Function Belonging to it
36. The Submodules of F°
37. Finite Commutative Semigroups
38. Numerical Semimodules
39. Investigation of the Kernel Functions "in the Little"
40. The Numerical Semimodules Attached to the Kernel Functions
41. The Kernel Functions of First Rank
42. The Maximum Condition in the Lattice of Kernel Functions
43. The Normals of a Kernel Function
44. Splitting Kernel Functions
45. The Kernel Functions of Second Order
46. The Kernel Functions of Second Dimension
47. Degenerate Kernel Functions
Chapter V. Equivalent Kernel Functions
48. Preparations for the Solution of the Isomorphism Problem
49. Submodules of F°, Equivalent Relative to F
50. Equivalent Kernel Functions
Appendix
51. The Case of Semigroups without a Unity Element
Index
Other Titles in the Series
- No. of pages: 368
- Language: English
- Published: January 1, 1965
- Imprint: Pergamon
- eBook ISBN: 9781483155944