Skip to main content
SpringerLink
Log in

Decomposition and complexity of finite semigroups

  • Survey Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

The Prime Decomposition Theorem for finite transformation semigroups of Krohn and Rhodes and the resultant theory of complexity of finite semigroups are the subjects of this survey. All concepts used are defined in the text, and proofs are given if they are accessible without a large amount of preparation. This article is an attempt to lead the reader to the essential ideas of the theory in the shortest possible time and effort.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Clifford, A. H. and G. B. Preston,The algebraic theory of semigroups, Amer. Math. Soc. 1 (1961).

  2. Krohn, K. and J. Rhodes,Algebraic theory of machines, I.Prime decomposition theorem for finite semigroups and machines, Trans. Amer. Math. Soc. 116 (1965), 450–464.

    Article  MATH  MathSciNet  Google Scholar 

  3. Krohn, K., J. Rhodes and B. Tilson,Lectures on the algebraic theory of finite semigroups and finite state machines, Chapters 1, 5–9 (Chapter 6 with M. A. Arbib), Algebraic Theory of Machines, Languages and Semigroups (M. A. Arbib, Editor), Academic Press (1968)

  4. Krohn, K., R. Mateosian and J. Rhodes,Complexity of ideals in finite semigroups and finite state machines, J. Math. Systems Theory 1 (1967), 59–66+erratum.

    Article  MATH  MathSciNet  Google Scholar 

  5. Krohn, K. and J. Rhodes,Complexity of finite semigroups, Ann. of Math. 88 (1968), 128–160.

    Article  MathSciNet  Google Scholar 

  6. Rhodes, J.,The Fundamental Lemma of Complexity for arbitrary finite semigroups, Bull. Amer. Math. Soc. 74 (1968), 1104–1109.

    Article  MATH  MathSciNet  Google Scholar 

  7. Rhodes, J.,A proof of the Fundamental Lemma of Complexity (Weak version)for arbitrary finite semigroups, J. Combinatorial Theory, Series A, 10 (1971), 22–73.

    Article  MATH  MathSciNet  Google Scholar 

  8. Rhodes, J.,A proof of the Fundamental Lemma of Complexity (strong version)for arbitrary finite semigroups, J. Combinatorial Theory (to appear).

  9. Rhodes, J.,Axioms for complexity for all finite semigroups, Adv. in Math. (to appear).

  10. Rhodes, J.,Algebraic theory of finite semigroups.Structure numbers and structure theorems for finite semigroups, Semigroups: Proceedings of a Symposium, Academic Press (1969).

  11. Rhodes, J.,A homomorphism theorem for finite semigroups, J. Math. Systems Theory, 1 (1967), 289–304.

    Article  MATH  MathSciNet  Google Scholar 

  12. Rhodes, J. and B. Tilson,Lower bounds for complexity of finite semigroups, J. Pure and Applied Algebra 1 (1971), 79–95.

    Article  MathSciNet  MATH  Google Scholar 

  13. Rhodes, J. and B. Tilson,Improved lower bounds for complexity of finite semigroups, submitted to J. of Pure and Applied Algebra.

  14. Stiffler, P.,Extension of the fundamental theorem of finite semigroups, Ph. D. thesis, Univ. of Calif., Berkeley (1970).

    Google Scholar 

  15. Tilson, B.,Complexity of-J class semigroups, Adv. in Math. (to appear).

  16. Rhodes, J.,Some results on finite semigroups, J. Algebra 4 (1966), 471–504.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Columbia University, 10027, New York, N. Y.

    Bret Tilson

Authors
  1. Bret Tilson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tilson, B. Decomposition and complexity of finite semigroups. Semigroup Forum 3, 189–250 (1971). https://doi.org/10.1007/BF02572961

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02572961

Keywords

Search

Navigation