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The loop problem for Rees matrix semigroups

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Abstract

We study the relationship between the loop problem of a semigroup, and that of a Rees matrix construction (with or without zero) over the semigroup. This allows us to characterize exactly those completely zero-simple semigroups for which the loop problem is context-free. We also establish some results concerning loop problems for subsemigroups and Rees quotients.

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References

  1. Ayik, H., Ruškuc, N.: Generators and relations of Rees matrix semigroups. Proc. Edinb. Math. Soc. 42, 481–495 (1999)

    MATH  Google Scholar 

  2. Belegradek, O.V.: Higman’s embedding theorem in a general setting and its application to existentially closed algebras. Notre Dame J. Form. Log. 37(4), 613–624 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belyaev, V.Y.: Imbeddability of recursively defined inverse semigroups in finitely presented semigroups. Sib. Mat. Zh. 25(2), 50–54 (1984)

    MathSciNet  Google Scholar 

  4. Boone, W.W., Higman, G.: An algebraic characterization of groups with soluble word problem. J. Aust. Math. Soc. 18, 41–53 (1974). Collection of articles dedicated to the memory of Hanna Neumann, IX

    MATH  MathSciNet  Google Scholar 

  5. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I. Am. Math. Soc., Providence (1961)

    MATH  Google Scholar 

  6. Dehn, M.: Über unendliche diskontinuierliche Gruppen. Math. Ann. 71(1), 116–144 (1911)

    Article  MathSciNet  Google Scholar 

  7. Descalço, L., Ruškuc, N.: On automatic Rees matrix semigroups. Commun. Algebra 30, 1207–1226 (2002)

    Article  MATH  Google Scholar 

  8. Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math. 81(3), 449–457 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  9. Elder, M., Kambites, M., Ostheimer, G.: On groups and counter automata. arXiv:math.RA/0611188 (2006)

  10. Herbst, T.: On a subclass of context-free groups. RAIRO Inf. Théor. Appl. 25(3), 255–272 (1991)

    MATH  MathSciNet  Google Scholar 

  11. Holt, D.F., Rees, S., Röver, C.E., Thomas, R.M.: Groups with context-free co-word problem. J. Lond. Math. Soc. (2) 71(3), 643–657 (2005)

    Article  MATH  Google Scholar 

  12. Hopcroft, J.E., Ullman, J.D.: Formal Languages and their Relation to Automata. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  13. Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon, Oxford (1995)

    MATH  Google Scholar 

  14. Kambites, M.: The loop problem for monoids and semigroups. Math. Proc. Camb. Philos. Soc. (to appear)

  15. Kambites, M.: Presentations for semigroups and semigroupoids. Int. J. Algebra Comput. 15, 291–308 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kambites, M.: Automatic semigroups and categories. Theor. Comput. Sci. 353, 272–290 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kambites, M.: Word problems recognisable by deterministic blind monoid automata. Theor. Comput. Sci. 362(1), 232–237 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kharlampovich, O.G., Sapir, M.V.: Algorithmic problems in varieties. Int. J. Algebra Comput. 5(4–5), 379–602 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kublanovskiĭ, S.I.: On varieties of associative algebras with local finiteness conditions. Algebra Analiz 9(4), 119–174 (1997)

    Google Scholar 

  20. Kukin, G.: The variety of all rings has Higman’s property. In: Algebra and Analysis, Irkutsk, 1989. Am. Math. Soc. Transl. Ser. 2, vol. 163, pp. 91–101. Am. Math. Soc., Providence (1995)

    Google Scholar 

  21. Lawson, M.V.: Rees matrix semigroups over semigroupoids and the structure of a class of abundant semigroups. Acta Sci. Math. 66, 517–540 (2000)

    MATH  MathSciNet  Google Scholar 

  22. Lawson, M.V.: Finite Automata. Chapman and Hall, London (2003)

    Google Scholar 

  23. McAlister, D.B.: Rees matrix covers for locally inverse semigroups. Trans. Am. Math. Soc. 227, 727–738 (1983)

    Article  MathSciNet  Google Scholar 

  24. Muller, D.E., Schupp, P.E.: Groups, the theory of ends, and context-free languages. J. Comput. Syst. Sci. 26(3), 295–310 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Murskiĭ, V.L.: Isomorphic imbeddability of a semigroup with an enumerable set of defining relations into a finitely presented semigroup. Mat. Zamet. 1, 217–224 (1967)

    Google Scholar 

  26. Petrich, M.: Embedding semigroups into idempotent generated ones. Monatsh. Math. 141(4), 315–322 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pin, J.-E.: Varieties of Formal Languages. North Oxford Academic (1986)

  28. Rees, D.: On semi-groups. Proc. Camb. Philos. Soc. 36, 387–400 (1940)

    MATH  MathSciNet  Google Scholar 

  29. Stewart, I.A., Thomas, R.M.: Formal languages and the word problem for groups. In: Groups St. Andrews 1997 in Bath, II. London Math. Soc. Lecture Note Ser., vol. 261, pp. 689–700. Cambridge Univ. Press, Cambridge (1999)

    Google Scholar 

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  1. School of Mathematics, University of Manchester, Manchester, M13 9PL, England

    Mark Kambites

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  1. Mark Kambites
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Correspondence to Mark Kambites.

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Communicated by Steve Pride

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Kambites, M. The loop problem for Rees matrix semigroups. Semigroup Forum 76, 204–216 (2008). https://doi.org/10.1007/s00233-007-9016-6

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