Skip to main content
SpringerLink
Log in

String Rewriting and Gröbner Bases — A General Approach to Monoid and Group Rings

  • Conference paper
Symbolic Rewriting Techniques

Part of the book series: Progress in Computer Science and Applied Logic ((PCS,volume 15))

Abstract

The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Gröbner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some non-commutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Gröbner bases for finitely generated right ideals is shown and a procedure to compute them is given.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Apel, Gröbnerbasen in nichtkommutativen Algebren und ihre Anwendung PhD Thesis. Leipzig. 1988.

    Google Scholar 

  2. J. Apel and W. Lassner. An Extension of Buchberg er’s Algorithm and Calculations in Enveloping Fields of Lie Algebras. Journal of Symbolic Computation(1988) 6. pp 361–370.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Apel. A Gröbner Approach to Involutive Bases. Journal of Symbolic Computation(1995) Vol. 19 No. 5. pp 441–457.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Avenhaus and K. Madlener. The Nielsen Reduction and P-Complete Problems in Free Groups. Theoretical Computer Science 32(1984). pp 61–76.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Avenhaus, K. Madlener, F. Otto. Groups Presented by Finite Two-Monadic Church-Rosser Thue Systems. Transactions of the American Mathematical Society. Vol. 297(1986). pp 427–443.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Bachmair and H. Ganzinger. Rewrite-Based Equational Theorem Proving With Selection Simplification. Journal of Symbolic Computation(1994) Vol. 4 No. 3. pp 1–31.

    MathSciNet  Google Scholar 

  7. L. Bachmair and H. Ganzinger. Buchberger’s algorithm: A constraint-based completion procedure. Proc. CCL′94. pp 285-301.

    Google Scholar 

  8. G. Bauer. Zur Darstellung von Monoiden durch konfluente Reduktionssysteme. PhD Thesis. Universität Kaiserslautern. 1981.

    Google Scholar 

  9. G. Baumslag, F. Cannonito and C. Miller III. Computable Algebra and Group Embeddings. Journal of Algebra 69(1981). pp 186–212.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Becker and V. Weispfenning. Gröbner Bases — A Computational Approach to Commutative Algebra. Springer Verlag(1992).

    Google Scholar 

  11. B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD Thesis. Universität Innsbruck. 1965.

    Google Scholar 

  12. B. Buchberger. A Critical-Pair Completion Algorithm for Finitely Generated Ideals in Rings. Proc. Logic and Machines: Decision Problems and Complexity. Springer LNCS 171. pp 137-161.

    Google Scholar 

  13. B. Buchberger. Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. N. K. Bose (ed). Multidimensional Systems Theory. Chapter 6. 1985. Dordrecht: Reidel. pp 184–232.

    Chapter  Google Scholar 

  14. B. Buchberger. Applications of Gröbner Bases in Non-Linear Computational Geometry. In: R. Janßen (ed.). Trends in Computer Algebra. Springer LNCS 296(1987). pp 52–80.

    Google Scholar 

  15. R. Bündgen. Simulating Buchberger’s Algorithm by a Knuth-Bendix Completion Procedure. Proc. RTA′91. pp 386-397.

    Google Scholar 

  16. R. Book and F. Otto. String-Rewriting Systems. Springer Verlag(1993).

    Google Scholar 

  17. D. Cox, J. Little and D. O’Shea. Ideals, Varieties, and Algorithms. Springer Verlag(1992).

    Google Scholar 

  18. R. Cremanns and F. Otto. Constructing Canonical Presentations for Subgroups of Context-Free Groups in Polynomial Time. Proc. ISSAC′94.

    Google Scholar 

  19. M. Dehn. Über unendliche diskontinuierliche Gruppen. Mathematische Annalen 71(1912). pp 116–144.

    Article  MathSciNet  MATH  Google Scholar 

  20. T. Deiß. Conditional Semi-Thue Systems for Presenting Monoids. Proc. STACS′92. pp 557-565.

    Google Scholar 

  21. D. Farkas, C. Feustel, E. Green. Synergy in the theories of Gröbner bases and path algebras. Canadian Journal of Mathematics. Vol. 45 Nr. 4(1993). pp 727–739.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Galligo. Some algorithmic questions on ideals of differential operators. Proc. EUROCAL ′85 II. LNCS 204 (1985). pp 413–421.

    MathSciNet  Google Scholar 

  23. K. O. Geddes, S. R. Czapor and G. Labahn. Algorithms for Computer Algebra. Kluwer Academic Publishers, Massachusettes (1992).

    Book  MATH  Google Scholar 

  24. R. Gilman. Presentations of Groups and Monoids. Journal of Algebra 57(1979). pp 544–554.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Hironaka. Resolution of singularities of an Algebraic Variety over a Field of Characteristic Zero. Annals of Mathematics 79(1964). pp 109–326.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Huet. Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems. Journal of the ACM 27(4)(1980). pp 797–821.

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Huet. A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm. Journal of Computer and System Science 23(1)(1981). pp 11–21.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Jantzen. On a Special Monoid with Single Defining Relation. Theoretical Computer Science 16(1981). pp 61–73.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Jantzen. A Note on a Special One-rule Semi-Thue System. Information Processing Letters 21(1985). pp 135–140.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Kandri-Rody and D. Kapur. An Algorithm for Computing the Gröbner Basis of a Polynomial Ideal over an Euclidean Ring. Technical Information Series General Electric Company Corporate Research and Development Schenectady, NY 12345. Dec. 1984.

    Google Scholar 

  31. A. Kandri-Rody and D. Kapur. Computing a Gröbner Basis of a Polynomial Ideal over an Euclidean domain. Journal of Symbolic Computation 6(1988). pp 37–57.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Kandri-Rody, D. Kapur and F. Winkler. Knuth-Bendix Procedure and Buchberger Algorithm — a Synthesis. Proc. ISSAC′89. pp 55-67.

    Google Scholar 

  33. M. I. Kargapolov and Ju. I. Merzljakov. Fundamentals of the Theory of Groups. Springer Verlag(1979).

    Google Scholar 

  34. D. Kapur and K. Madlener. Private communication.

    Google Scholar 

  35. D. Kapur and P. Narendran. Constructing a Gröbner Basis for a Polynomial Ring. Proc. Combinatorial Algorithms in Algebraic Structures. Otzenhausen(1985). Eds J. Avenhaus, K. Madlener. Universität Kaiserslautern.

    Google Scholar 

  36. D. Kapur and P. Narendran. A Finite Thue System with Decidable Word Problem and Without Equivalent Finite Canonical System. Theoretical Computer Science 35(1985). pp 337–344.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Kandri-Rody and V. Weispfenning. Non-Commutative Gröbner Bases in Algebras of Solvable Type. Journal of Symbolic Computation 9(1990). pp 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  38. B. J. Keller. Alternatives in Implementing Noncommutative Gröbner Basis Systems. See this volume.

    Google Scholar 

  39. D. Knuth and P. Bendix. Simple Word Problems in Universal Algebras. J. Leech (editor). Computaional Problems in Abstract Algebra. Pergamon Press, Oxford. 1970. pp 263–297.

    Google Scholar 

  40. H. Kredel. Solvable Polynomial Rings. Verlag Shaker, Aachen. 1993.

    MATH  Google Scholar 

  41. N. Kuhn and K. Madlener. A Method for Enumerating Cosets of a Group Presented by a Canonical System. Proc. ISSAC′89. pp 338-350.

    Google Scholar 

  42. N. Kuhn, K. Madlener and F. Otto. Computing Presentations for Subgroups of Polycyclic Groups and of Context-Free Groups. Applicable Algebra in Engineering, Communication and Computing 5(1994). pp 287–316.

    Article  MathSciNet  MATH  Google Scholar 

  43. W. Lassner. Symbol Representations of Noncommutative Algebras. Proc. EUROCAL′85. Springer LNCS 204. pp 99-115.

    Google Scholar 

  44. M. Lauer. Kanonische Repräsentanten für die Restklassen nach einem Polynomideal. Diplomarbeit. Universität Kaiserslautern. 1976.

    Google Scholar 

  45. P. Le Chenadec. Canoniacl Forms in Finitely Presented Algebras. Pitman/Wiley. London. 1986.

    Google Scholar 

  46. R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer Verlag(1977).

    Google Scholar 

  47. K. Madlener. Existence and Construction of Gröbner Bases for Ideals in Reduction Rings. Working paper. 1986.

    Google Scholar 

  48. K. Madlener, P. Narendran, F. Otto and L. Zhang. On Weakly Confluent Monadic String-Rewriting Systems. Theoretical Computer Science 113(1993). pp 119–165.

    Article  MathSciNet  MATH  Google Scholar 

  49. K. Madlener and F. Otto. About the Descriptive Power of Certain Classes of Finite String-Rewriting Systems. Theoretical Computer Science 67(1989). pp 143–172.

    Article  MathSciNet  MATH  Google Scholar 

  50. K. Madlener and F. Otto. Some Undecidability Results for Finitely Generated Thue Congruences on a Two-Letter Alphabet. E. Schock (ed.). Beiträge zur Angewandten Analysis und Informatik, Helmut Brakhage zu Ehren. Verlag Shaker, Aachen. 1994. pp 248–261.

    Google Scholar 

  51. K. Madlener and B. Reinert. On Gröbner Bases in Monoid and Group Rings. SEKI Report SR-93-08. Universität Kaiserslautern.

    Google Scholar 

  52. K. Madlener and B. Reinert. Computing Gröbner Bases in Monoid and Group Rings. Proc. ISSAC′93. pp 254-263.

    Google Scholar 

  53. K. Madlener and B. Reinert. On Gröbner Bases for Two-Sided Ideals in Nilpotent Group Rings. SEKI Report SR-95-01. Universität Kaiserslautern.

    Google Scholar 

  54. K. Madlener and B. Reinert. A Generalization of Gröbner Bases Algorithms to Nilpotent Group Rings. Applicable Algebra in Engineering, Communication and Computing Vol. 8 No. 2(1997). pp 103–123.

    Article  MathSciNet  MATH  Google Scholar 

  55. K. Madlener and B. Reinert. A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings. Journal of Symbolic Computation. To appear.

    Google Scholar 

  56. K. Madlener and B. Reinert. Congruemces in Monoids and Ideals in Monoid Rings. Technical Report. Universität Kaiserslautern. 1997.

    Google Scholar 

  57. W. Magnus, A. Karrass and D. Solitar. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Dover Publications. New York. 1976.

    MATH  Google Scholar 

  58. C. Marché. Normalized Rewriting — Application to Ground Completion and Standard Bases. In: H. Comon, J.-P. Jouannaud (Eds.). Term Rewriting, Springer. 1993. pp 154–169.

    Google Scholar 

  59. F. Mora. Gröbner Bases for Non-Commutative Polynomial Rings. Proc. AAECC-3(1985). Springer LNCS 229. pp 353-362.

    Google Scholar 

  60. T. Mora. Gröbner bases for non-commutative algebras. Proc. ISSAC′88. LNCS 358(1989). pp 150-161.

    Google Scholar 

  61. T. Mora. An Introduction to Commutative and Non-Commutative Gröbner Bases. Theoretical Computer Science 134(1994). pp 131–173.

    Article  MathSciNet  MATH  Google Scholar 

  62. J. Nielsen. Om Regning med ikke kommutative Faktoren og dens Anvendelse i Gruppeteorien. Mat. Tidsskr. B.(1921). pp 77–94.

    Google Scholar 

  63. C. Ó’Dúnlaing. Undecidable Questions Related to Church-Rosser Thue Systems. Theoretical Computer Science 23(1983). pp 339–345.

    Article  MathSciNet  Google Scholar 

  64. F. Otto. On Deciding the Confluence of a Finite String-Rewriting System on a Given Congruence Class. Journal of Computer and System Science 35(1987). pp 285–310.

    Article  MathSciNet  MATH  Google Scholar 

  65. F. Otto and L. Zhang. Decision Problems for Finite Special String-Rewiting Systems that are Confluent on some Congruence Class. Acta Informatica 28(1991). pp 477–510.

    Article  MathSciNet  MATH  Google Scholar 

  66. L. Pan. On the Gröbner Bases of Ideals in Polynomial Rings over a Prinicipal Ideal Domain. University of California, Santa Barbara. Department of Mathematics. Internal Manuscript. 1985.

    Google Scholar 

  67. M. Pesch. Two-sided Gröbner Bases in Iterated Ore Extensions. See this volume.

    Google Scholar 

  68. B. Reinert. Gröbner Bases in Monoid and Group Rings. PhD Thesis. Universität Kaiserslautern. 1995.

    Google Scholar 

  69. B. Reinert. Introducing Reduction to Poly cyclic Group Rings — A Comparison of Methods. Reports on Computer Algebra No 9. Centre of Computer Algebra. Universität Kaiserslautern. 1996.

    Google Scholar 

  70. A. Rosenmann. An Algorithm for Constructing Gröbner and Free Schreier Bases in Free Group Algebras. Journal of Symbolic Computation 16(1993). pp 523–549.

    Article  MathSciNet  MATH  Google Scholar 

  71. A. Sattler-Klein. Divergence Phenomena During Completion. Proc. RTA′91. pp 374-385.

    Google Scholar 

  72. A. Sattler-Klein. A Systematic Study of Infinite Canonial Systems generated by Knuthe-Bendix Completion and Related Problems. PhD Thesis. Universität Kaiserslautern. 1996.

    Google Scholar 

  73. C. Sims. Verifying Nilpotence. Journal of Symbolic Computation 3(1987). pp 231–247.

    Article  MathSciNet  MATH  Google Scholar 

  74. C. Sims. Implementing the Baumslag-Cannonito-Miller Polycyclic Quotient Algorithm. Journal of Symbolic Computation 9(1990). pp 707–723.

    Article  MathSciNet  MATH  Google Scholar 

  75. C. Sims. Computation with finitely presented groups. Cambridge University Press 1994.

    Google Scholar 

  76. C. Squier. Word Problems and a Homological Finiteness Condition for Monoids. Journal of Pure Applied Algebra 49(1987). pp 201–217.

    Article  MathSciNet  MATH  Google Scholar 

  77. S. Stifter. Computation of Gröbner Bases over the Integers and in General Reduction Rings. Diplomarbeit, Johannes Kepler Universität Linz. 1985.

    Google Scholar 

  78. S. Stifter. A generalization of Reduction Rings. Journal of Symbolic Computation 4(1987). pp 351–364.

    Article  MathSciNet  MATH  Google Scholar 

  79. T. Stokes. Gröbner Bases in Exterior Algebras. Journal of Automated Reasoning 6(1990). pp 233–250.

    Article  MathSciNet  MATH  Google Scholar 

  80. V. Weispfenning. Gröbner Basis for Polynomial Ideals over Commutative Regular Rings. Proc. EUROCAL′87. Springer LNCS 378. pp 336-347.

    Google Scholar 

  81. V. Weispfenning. Finite Gröbner Bases in Non-Noetherian Skew Polynomial Rings. Proc. ISSAC′92. pp 329-334.

    Google Scholar 

  82. D. Wißmann. Applying Rewriting Techniques to Groups with Power-Commutation-Presentations. Proc. ISSAC′88. pp 378-389.

    Google Scholar 

  83. D. Wißmann. Anwendung von Rewriting-Techniken in polyzyklischen Gruppen. PhD Thesis. Universität Kaiserslautern. 1989.

    Google Scholar 

  84. A. Zharkov and Yu. Blinkov. Involution Approach to Solving Systems of Algebraic Equations. Proc. IMACS′93. pp 11-16.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Kaiserslautern, 67663, Kaiserslautern, Germany

    Klaus Madlener & Birgit Reinert

Authors
  1. Klaus Madlener
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Birgit Reinert
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. INRIA, 2004, Route des lucioles, BP 93, 06902, Sophia Antipolis Cedex, France

    Manuel Bronstein

  2. FMI Universität Passau, 94030, Passau, Germany

    Volker Weispfenning

  3. IBM Deutschland Informationssysteme GmbH, Postfach 10 30 68, 69020, Heidelberg, Germany

    Johannes Grabmeier

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Basel AG

About this paper

Cite this paper

Madlener, K., Reinert, B. (1998). String Rewriting and Gröbner Bases — A General Approach to Monoid and Group Rings. In: Bronstein, M., Weispfenning, V., Grabmeier, J. (eds) Symbolic Rewriting Techniques. Progress in Computer Science and Applied Logic, vol 15. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8800-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8800-4_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9779-2

  • Online ISBN: 978-3-0348-8800-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation