Skip to main content

A Noetherian and confluent rewrite system for idempotent semigroups

Semigroup Forum volume 25, pages 83–110 (1982)Cite this article

Abstract

Let B be a semigroup with the additional relation

$$\begin{gathered} xx \Rightarrow x \hfill \\ xyz \Rightarrow xz if x \mathop {CI}\limits_ = z and xy\mathop {CI}\limits_ = z \hfill \\ \end{gathered} $$

B is called aband or anidempotent semigroup [3].

It is shown in this paper that the replacement rules (rewrites) resulting from the axiom of idempotence:

$$\forall w \in B.ww = w$$

can be replaced by theNoetherian, confluent, conditional rewrites (i. e. a terminating replacement system having the Church-Rosser-Property):

$$\begin{gathered} xx \Rightarrow x \hfill \\ x \Rightarrow xx \hfill \\ \end{gathered} $$

These rewrites are used to obtain a unique normal form for words in B and hence are the basis for a decision procedure for word equality in B.

The proof techniques are based uponterm rewriting systems [7] rather than the usual algebraic approach. Alternative and simpler proofs of a result reported earlier by Green and Rees [4] and Gerhardt [6] have been obtained.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. D. Brand, J. Darringer, J. Joyner “Completeness of Conditional Reductions”; Proc. of the Workshop on Automated Deduction, 1979

  2. T.C. Brown “On the Finiteness of Semigroups in which xr=x″; Proc. Cambridge Phil. Soc. 60, 1964

  3. A.H. Clifford, G.B. Preston “The Algebraic Theory of Semigroups”; American Math. Society, 1961

  4. J.H. Green, D. Rees “On Semigroups in which xr=x″, Proc. Cambridge Phil. Soc. 48, 1952

  5. J.V. Guttag, E. Horowith, D.R. Musser “Abstract Data Types and Software Validation”; Com. of the ACM, vol. 21, no. 12, 1978

  6. J.A. Gerhardt “The Lattice of Equational Classes of Idempotent Semigroups”; J. of Algebra, 15, 1970

  7. G. Huet, D. Oppen “Equations and Rewrite Rules: A Survey”; in: ‘Formal Language Theory: Perspectives and Open Problems’; R.V. Book (ed), Academic Press, 1980

  8. J.M. Hullot “A Catalogue of Canonical Term Rewriting Systems”; Technical Report CSL-113, Stanford Research Institute, 1980

  9. J.M. Howie “An Introduction to Semigroup Theory”, Academic Press, 1976

  10. M.A. Harrison “Introduction to Formal Language Theory”, Addison Wesley, 1978

  11. G. Huet “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems”; JACM, vol 27, no. 4, 1980

  12. D. Knuth, P. Bendix “Simple Word Problems in Universal Algebras”, in: Computational Problems in Abstract Algebra, (ed) J. Leech, Pergamon Press, 1970

  13. D.S. Lankford, A.M. Ballantyne “Decision Procedure for Simple Equational Theories with Permutative Axioms”, University of Texas, Report ATP-37

  14. D. Loveland “Automated Theorem Proving”, North Holland Publ. Comp., 1978

  15. D. McLean “Idempotent Semigroups”, Americ. Math. Mon. 61, 1954

  16. P. Raulefs, J. Siekmann, P. Szabó, E. Univericht “A Short Survey on the State of the Art in Matching and Unification Problems”, Bulletin of EATC, Oct. 1978

  17. J. Siekmann, P. Szabó “Unification in Idempotent Semigroups”; Universität Karlsruhe, Institut für Informatik I (in preparation)

  18. J. Siekmann “Unification and Matching Problems”, Universität Karlsruhe, Institut für Informatik I, 1978

  19. J. Siekmann, P. Szabó “Universal Unification”, Universität Karlsruhe, Institut für Informatik I, 1981

  20. P. Szabó “Unifikations theorie erster Ordnung”, Universität Karlsruhe, Ph.D. (in German)

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik I, Universität Karlsruhe, Postfach 6380, 7500, Karlsruhe 1, Federal Republic of Germany

    J. Siekmann & P. Szabó

Authors
  1. J. Siekmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Szabó
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by K. Keimel

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Siekmann, J., Szabó, P. A Noetherian and confluent rewrite system for idempotent semigroups. Semigroup Forum 25, 83–110 (1982). https://doi.org/10.1007/BF02573590

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02573590

Keywords

  • Relative Length
  • Equational Theory
  • Unification Algorithm
  • Replacement Rule
  • Abstract Data Type