# The Theory of Finitely Generated Commutative Semigroups

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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.

## Table of Contents

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

## Product details

- No. of pages: 368
- Language: English
- Copyright: © Pergamon 1965
- Published: January 1, 1965
- Imprint: Pergamon
- eBook ISBN: 9781483155944

## About the Editors

### I. N. Sneddon

### M. Stark

### K. A. H. Gravett

## About the Author

### L. Rédei

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