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On deciding whether a monoid is a free monoid or is a group

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In general it is undecidable whether or not the monoid described by a given finite presentation is a free monoid or a group. However, these two decision problems are reducible to a very restricted form of the uniform word problem. So whenever a class of presentations is considered for which this restricted form of the uniform word problem is decidable, then the above decision problems become decidable. This holds in particular for the class of all presentations involving finite string-rewriting systems that are noetherian and confluent.

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References

  1. Book, R.V.: When is a monoid a group? The Church-Rosser case is tractable. Theor. Comput. Sci. 18, 325–331 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Book, R.V.: Decidable sentences of Church-Rosser congruences. Theor. Comput. Sci. 24, 301–312 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  3. Book, R.V.: Thue systems as rewriting systems. In: Rewriting techniques and applications. (J.P. Jouannaud, ed). Lect. Notes Comput. Sci. 202, pp. 63–94. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  4. Lallement, G.: Semigroups and combinatorial applications. New York, Chichester, Brisbane, Toronto: John Wiley 1979

    Google Scholar 

  5. Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory, 2nd. rev. ed., New York: Dover 1976

    Google Scholar 

  6. Markov, A.A.: Impossibility of algorithms for recognizing some properties of associative systems. Dokl. Akad. Nauk SSSR 77, 953–956 (1951)

    MATH  Google Scholar 

  7. Mostowski, A.: (Review of Reference 6) J. Symb. Logic 17, 151–152 (1952)

    MathSciNet  Google Scholar 

  8. Narendran, P., Otto, F.: Complexity results on the conjugacy problem for monoids. Theor. Comput. Sci. 35, 227–243 (1985)

    Article  MathSciNet  Google Scholar 

  9. Otto, F.: Some undecidability results for non-monadic Church-Rosser Thue systems. Theor. Comput. Sci. 33, 261–278 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Otto, F.: Church-Rosser Thue systems that present free monoids. SIAM J. Comput. (To appear)

  11. Tietze, H.: Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten. Monatsh. Math. Phys. 19, 1–118 (1908)

    MATH  Google Scholar 

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  1. Fachbereich Informatik, Universität Kaiserslautern, Postfach 3049, D-6750, Kaiserslautern, Germany

    F. Otto

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  1. F. Otto
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Otto, F. On deciding whether a monoid is a free monoid or is a group. Acta Informatica 23, 99–110 (1986). https://doi.org/10.1007/BF00268077

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  • DOI: https://doi.org/10.1007/BF00268077

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