Semigroup Forum

, Volume 25, Issue 1, pp 83–110 | Cite as

A Noetherian and confluent rewrite system for idempotent semigroups

  • J. Siekmann
  • P. Szabó
Research Article


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.


Relative Length Equational Theory Unification Algorithm Replacement Rule Abstract Data Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Brand, J. Darringer, J. Joyner “Completeness of Conditional Reductions”; Proc. of the Workshop on Automated Deduction, 1979Google Scholar
  2. [2]
    T.C. Brown “On the Finiteness of Semigroups in which xr=x″; Proc. Cambridge Phil. Soc. 60, 1964Google Scholar
  3. [3]
    A.H. Clifford, G.B. Preston “The Algebraic Theory of Semigroups”; American Math. Society, 1961Google Scholar
  4. [4]
    J.H. Green, D. Rees “On Semigroups in which xr=x″, Proc. Cambridge Phil. Soc. 48, 1952Google Scholar
  5. [5]
    J.V. Guttag, E. Horowith, D.R. Musser “Abstract Data Types and Software Validation”; Com. of the ACM, vol. 21, no. 12, 1978Google Scholar
  6. [6]
    J.A. Gerhardt “The Lattice of Equational Classes of Idempotent Semigroups”; J. of Algebra, 15, 1970Google Scholar
  7. [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, 1980Google Scholar
  8. [8]
    J.M. Hullot “A Catalogue of Canonical Term Rewriting Systems”; Technical Report CSL-113, Stanford Research Institute, 1980Google Scholar
  9. [9]
    J.M. Howie “An Introduction to Semigroup Theory”, Academic Press, 1976Google Scholar
  10. [10]
    M.A. Harrison “Introduction to Formal Language Theory”, Addison Wesley, 1978Google Scholar
  11. [11]
    G. Huet “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems”; JACM, vol 27, no. 4, 1980Google Scholar
  12. [12]
    D. Knuth, P. Bendix “Simple Word Problems in Universal Algebras”, in: Computational Problems in Abstract Algebra, (ed) J. Leech, Pergamon Press, 1970Google Scholar
  13. [13]
    D.S. Lankford, A.M. Ballantyne “Decision Procedure for Simple Equational Theories with Permutative Axioms”, University of Texas, Report ATP-37Google Scholar
  14. [14]
    D. Loveland “Automated Theorem Proving”, North Holland Publ. Comp., 1978Google Scholar
  15. [15]
    D. McLean “Idempotent Semigroups”, Americ. Math. Mon. 61, 1954Google Scholar
  16. [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. 1978Google Scholar
  17. [17]
    J. Siekmann, P. Szabó “Unification in Idempotent Semigroups”; Universität Karlsruhe, Institut für Informatik I (in preparation)Google Scholar
  18. [18]
    J. Siekmann “Unification and Matching Problems”, Universität Karlsruhe, Institut für Informatik I, 1978Google Scholar
  19. [19]
    J. Siekmann, P. Szabó “Universal Unification”, Universität Karlsruhe, Institut für Informatik I, 1981Google Scholar
  20. [20]
    P. Szabó “Unifikations theorie erster Ordnung”, Universität Karlsruhe, Ph.D. (in German)Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • J. Siekmann
    • 1
  • P. Szabó
    • 1
  1. 1.Institut für Informatik IUniversität KarlsruheKarlsruhe 1Federal Republic of Germany

Personalised recommendations