The Theory of Finitely Generated Commutative Semigroups
International Series of Monographs In: Pure and Applied Mathematics
Published on 10. July 2014
368 pages
978-1-4831-5594-4 (ISBN)
Description
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.
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Content
PrefaceIntroductionChapter 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 TheoremChapter 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 VChapter 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 FChapter 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 FunctionsChapter V. Equivalent Kernel Functions 48. Preparations for the Solution of the Isomorphism Problem 49. Submodules of F°, Equivalent Relative to F 50. Equivalent Kernel FunctionsAppendix 51. The Case of Semigroups without a Unity ElementIndexOther Titles in the Series
