Skip to main content
SpringerLink
Log in

On some algorithmic problems for groups and monoids

  • Conference paper
Book cover Rewriting Techniques and Applications (RTA 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 690))

Included in the following conference series:

  • 197 Accesses

Abstract

In 1912 Max Dehn formulated three main algorithmic problems for groups presented by defining relations: Word problem, Conjugacy problem and Isomorphism problem. Two years later A. Thue formulated the Word problem for semigroups presented by defining relations (Thue systems).

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

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. Adian S.I. (1955). Algorithmic unsolvability of the problem of recognizing certain properties of groups. Dokl. Akad Nauk SSSR 103, 533–535.

    MathSciNet  Google Scholar 

  2. Adian S.I. (1957). Unsolvability of certain algorithmic problems of group theory. Trudy Moskov. Mat. Obshch. 6, 231–298.

    Google Scholar 

  3. Adian S.I. (1957). Finitely presented groups and algorithms. Dokl. Akad Nauk SSSR 117, 9–12.

    MathSciNet  Google Scholar 

  4. Adian S.I. (1966). Defining relations and algorithmic problems for groups and semigroups. Proc. Steklov Inst. Math. 85. (English version published by the American Mathematical Society, 1967).

    Google Scholar 

  5. Adian S.I. (1976). Word transformations in a semigroup that is given by a system of defining relations. Algebra i Logika 15, 611–621; English transl. in Algebra and Logic 15 (1976).

    MathSciNet  Google Scholar 

  6. Adian S.I. and Oganesian G.U. (1978). On the word and divisibility problems in semigroups with a single defining relation. Izv. Akad. Nauk SSSR (Ser. Mat.) 42, 219–225; English. transl. in Math. USSR Izv. 12 (1978).

    MathSciNet  Google Scholar 

  7. Adian S.I. and Oganesian G.U. (1987). On the word and divisibility problems in semigroups with one defining relation. Mat. Zametki 41, 412–421; English transl. in Math. Notes 41 (1987).

    MathSciNet  Google Scholar 

  8. Adian S.I. and Makanin G.S. (1984). Investigations on algorithmic questions of algebra. Proc. Steklov Inst. Math. 168, 207–226; English transl. in Proc. Steklov Inst. Math. 3 (1986).

    Google Scholar 

  9. Book R.V. (1987) Thue Systems as Rewriting Systems. J. Symbolic Computation 3, 39–68.

    Article  MATH  MathSciNet  Google Scholar 

  10. Boone W.W. (1959). The word problem. Ann. of Math. (2) 70, 207–265.

    Article  MATH  MathSciNet  Google Scholar 

  11. Borisov V.V. (1969). Simple examples of groups with unsolvable word problem. Mat. Zametki 6, 521–532; English transl. in Math. Notes 6 (1969).

    MATH  MathSciNet  Google Scholar 

  12. Collins D.J. (1969) Recursively enumerable degrees and the cojugacy problem. Acta Math. 122, 115–160.

    Article  MATH  MathSciNet  Google Scholar 

  13. Dehn M. (1912). Ãœber unendliche diskontinuierliche Gruppen. Math. Ann. 71.

    Google Scholar 

  14. Durnev V.G. (1973). Positive theory of a free semigroup. Dokl. Akad Nauk SSSR 211, 772–774; English transl. in Soviet Math. Dokl. 14 (1973).

    MathSciNet  Google Scholar 

  15. Fridman A.A. (1962). Turing degrees of the word problem in finitely presented groups. Dokl. Akad Nauk SSSR 147, 805–808.

    MATH  MathSciNet  Google Scholar 

  16. Gol'berg A.I. (1978). On the impossibility of strenghtening certain results of Greendlinger and Lyndon. Uspekhi Mat. Nauk 33, 201–202; English transl. in Russian Math. Surveys 33 (1978).

    MATH  MathSciNet  Google Scholar 

  17. Howie J., Pride S. (1986). The word problem for one-relator semigroups. Math. Proc. of the Cambridge Phil. Soc. 99, 33–44.

    Article  MATH  MathSciNet  Google Scholar 

  18. Lentin A. (1972). Equations dans les monoides libres. Gauthier-Villars and Mouton, Paris.

    MATH  Google Scholar 

  19. Lyndon R. C. (1960). Equations in free groups. Trans. Amer. Math. Soc. 96, 445–457.

    Article  MATH  MathSciNet  Google Scholar 

  20. Lyndon R. C. (1966). On Dehn algorithm. Math. Ann. 166, 208–228.

    Article  MATH  MathSciNet  Google Scholar 

  21. Magnus W. (1932). Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106, 295–307.

    Article  MathSciNet  Google Scholar 

  22. Makanin G.S. (1977). The problem of solvability of equations in a free semigroups. Mat. Sb. 103 (145), 147–236; English transl. in Math. USSR Sb. 32 (1977).

    MathSciNet  Google Scholar 

  23. Makanin G.S. (1982). Eguations in a free group. Izv. Akad. Nauk SSSR (Ser. Mat.) 46, 1199–1273; English. transl. in Math. USSR Izv. 21 (1983).

    MATH  MathSciNet  Google Scholar 

  24. Makanin G.S. (1984). Decidability of the universal and positive theories in a free groups. Izv. Akad. Nauk SSSR (Ser. Mat.) 48, 735–749; English. transl. in Math. USSR Izv. 25 (1985).

    MathSciNet  Google Scholar 

  25. Marchenkov S.S. (1982). Undecidability of the positive AE-theory of a free semigroup. Sibirsk. Mat. Zh. 23, 196–198.

    MATH  MathSciNet  Google Scholar 

  26. Markov A.A. (1947). On the impossibility of certain algorithms in the theory of associative systems. Dokl. Akad Nauk SSSR 55, 683–586.

    Google Scholar 

  27. Markov A.A. (1954). The theory of Algorithms. Trudy Mat. Inst. Steklov 42; English transl., Israel Program Sci. Transl., Jerusalem (1961).

    Google Scholar 

  28. Matiyasevich Yu.V. (1967). Simple examples of undecidable associative calculi. Dokl. Akad Nauk SSSR 173, 1264–1266; English transl. in Soviet Math. Dokl. 8 (1967).

    MathSciNet  Google Scholar 

  29. Novikov P.S. (1955). On the algotithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44; English transl. in Amer. Math. Soc. Transl. (2) 9 (1958).

    Google Scholar 

  30. Oganesian G.U. (1982). On semigroups with one relation and semigroups without cycles. Izv. Akad. Nauk SSSR (Ser. Mat.) 46, 88–94; English. transl. in Math. USSR Izv. 20 (1983).

    MathSciNet  Google Scholar 

  31. Osipova V.A. (1968). On the word problem for finitely presented semigroups. Dokl. Akad Nauk SSSR 178, 1017–1020; English transl. in Soviet Math. Dokl. 9 (1968).

    MathSciNet  Google Scholar 

  32. Post E.L. (1947). Recursive unsolvability of a problem of Thue. J. Symbolic Logic 12, 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  33. Quine W.V. (1946). Concatenation as a basis for arithmetic. J. Symbolic Logic 11, 105–114.

    Article  MATH  MathSciNet  Google Scholar 

  34. Rabin M.O. (1958). Recursive unsolvavility of group theoretic problems. Ann. of Math. (2) 67, 172–194.

    Article  MATH  MathSciNet  Google Scholar 

  35. Razborov A.A. (1984). On systems of equations in a free groups. Izv. Akad. Nauk SSSR (Ser. Mat.) 48, 779–832; English. transl. in Math. USSR Izv. 25 (1985).

    MathSciNet  Google Scholar 

  36. Sarkisian O.A. (1979). Some relations between the word and divisibility problems in groups and semigroups. Izv. Akad. Nauk SSSR (Ser. Mat.) 43, 909–921; English. transl. in Math. USSR Izv. 15 (1980).

    MathSciNet  Google Scholar 

  37. Thue A. (1914). Probleme über Verbandenlungen von Zeichenreihen nach gegebenen Regeln. Kristiana Videnskapsselkapets Skr. I: Mat.-Naturvid. Kl. 10. (Reprinted in his Selected mathematical papers, Universitetsforlaget, Oslo (1977), 493–524).

    Google Scholar 

  38. Taimanov A.D. and Khmelevskii Yu.I. (1980). Decidability of the universal theory of a free semigroup. Sibirsk. Mat. Zh. 21, 228–230.

    MathSciNet  Google Scholar 

  39. Tartakovskii V.A. (1947). On the word problem for certain types of groups. Dokl. Akad Nauk SSSR 58, 1909–1910 (Russian).

    MathSciNet  Google Scholar 

  40. Tartakovskii V.A. (1949). Solution of the word problem for groups with a k-reduced basis for k>6. Izv. Akad. Nauk SSSR (Ser. Mat.) 13, 483–494; English transl. in Amer. Math. Soc. Transl. (1) 1 (1962).

    MathSciNet  Google Scholar 

  41. Tseitin G.S. (1956). Associative system with unsolvable word problem. Dokl. Akad Nauk SSSR 107, 370–371. (The full proof in Trudy Mat. Inst. Steklov 52 (1958), 172–189).

    MathSciNet  Google Scholar 

  42. Zhang L. (1991). Conjugacy in Special Monoids. J. of Algebra 143, 487–497.

    Article  MATH  Google Scholar 

  43. Zhang L. (1992). Applying rewriting methods to special monoids. Math. Proc. Cambridge Phil. Soc. 112, 495–505.

    Article  MATH  Google Scholar 

  44. Zhang L. (1992). A short proof of a theorem of Adjan. Proc. of the Amer. Math. Soc. 116, 1–3.

    Article  MATH  Google Scholar 

  45. Zhang L. (1992). On the conjugacy problem for one-relator monoids with elements of finite order. International J. of Algebra and Computations. 2, 209–220.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Vavilov str. 42, 117966, Moscow, Russia

    Sergei I. Adian

Authors
  1. Sergei I. Adian
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. INRIA Lorraine and CRIN, 615 Rue du Jardin Botanique, F-54602, Villers les Nancy Cedex, France

    Claude Kirchner

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Adian, S.I. (1993). On some algorithmic problems for groups and monoids. In: Kirchner, C. (eds) Rewriting Techniques and Applications. RTA 1993. Lecture Notes in Computer Science, vol 690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21551-7_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21551-7_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56868-1

  • Online ISBN: 978-3-662-21551-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation