The Theory of Finitely Generated Commutative Semigroups

The Theory of Finitely Generated Commutative Semigroups

1st Edition - January 1, 1965

Write a review

  • Editors: I. N. Sneddon, M. Stark, K. A. H. Gravett
  • eBook ISBN: 9781483155944

Purchase options

Purchase options
DRM-free (PDF)
Sales tax will be calculated at check-out

Institutional Subscription

Free Global Shipping
No minimum order


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


    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


    51. The Case of Semigroups without a Unity Element


    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

Ratings and Reviews

Write a review

There are currently no reviews for "The Theory of Finitely Generated Commutative Semigroups"