Skip to main content
SpringerLink
Log in

Fixed points of endomorphisms of trace monoids

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. If the dependence alphabet is a transitive forest, it is proved that the set of regular fixed points of the (Scott) continuous extension of an endomorphism to real traces is Ω-rational for every endomorphism if and only if the monoid is a free product of free commutative monoids.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aalbersberg, I.J., Hoogeboom, H.J.: Characterizations of the decidability of some problems for regular trace languages. Math. Syst. Theory 22, 1–19 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berstel, J.: Transductions and Context-Free Languages. Teubner, Stuttgart (1979)

    Book  MATH  Google Scholar 

  3. Bonizzoni, P., Mauri, G., Pighizzini, G.: About infinite traces. In: Diekert, V. (ed.) Proceedings of the ASMICS Workshop on Partially Commutative Monoids, pp. 1–10 (1990). Technische Universität München. Tech. Rep. TUM-I 9002

    Google Scholar 

  4. Cartier, P., Foata, D.: Problèmes Combinatoires de Commutation et Réarrangements. Lecture Notes in Mathematics, vol. 85. Springer, Berlin (1969)

    MATH  Google Scholar 

  5. Cassaigne, J., Silva, P.V.: Infinite periodic points of endomorphisms over special confluent rewriting systems. Ann. Inst. Fourier 59(2), 769–810 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  7. Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Am. J. Math. 35(4), 413–422 (1913)

    Article  MATH  Google Scholar 

  8. Diekert, V.: Combinatorics on Traces. Lecture Notes in Computer Science, vol. 454. Springer, Berlin (1990)

    MATH  Google Scholar 

  9. Diekert, V., Rozenberg, G. (eds.): The Book of Traces. World Scientific, Singapore (1995)

    Google Scholar 

  10. Gastin, P.: Infinite traces. In: Semantics of Systems of Concurrent Processes, Proceedings of the LITP Spring School on Theoretical Computer Science, La Roche Posay, France. LNCS, vol. 469, pp. 277–308. Springer, Berlin (1990)

    Chapter  Google Scholar 

  11. Kwiatkowska, M.Z.: A metric for traces. Inf. Process. Lett. 35, 129–135 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kwiatkowska, M.Z.: On the domain of traces and sequential composition. In: Proc. International Joint Conference on Theory and Practice of Software Development (TAPSOFT’91), pp. 42–56 (1991)

    Google Scholar 

  13. Lohrey, M., Steinberg, B.: The submonoid and rational subset membership problems for graph groups. J. Algebra 320(2), 728–755 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Metaftsis, V., Raptis, E.: On the profinite topology of right-angled Artin groups. J. Algebra 320(3), 1174–1181 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Perrin, D., Pin, J.-E.: Infinite Words: Automata, Semigroups, Logic and games. Pure and Applied Mathematics Series, vol. 141. Elsevier Academic, Amsterdam (2004)

    Google Scholar 

  16. Rodaro, E., Silva, P.V.: Fixed points of endomorphisms of trace monoids. Preprint. CMUP 2012-32. arXiv:1211.4517

  17. Rodaro, E., Silva, P.V., Sykiotis, M.: Fixed points of endomorphisms of graph groups. J. Group Theory 16(4), 573–583 (2013). doi:10.1515/jgt-2012-0047

    MATH  MathSciNet  Google Scholar 

  18. Silva, P.V.: Fixed points of endomorphisms over special confluent rewriting systems. Monatshefte Math. 161(4), 417–447 (2010)

    Article  MATH  Google Scholar 

  19. Silva, P.V.: Fixed points of endomorphisms of certain free products. RAIRO Theor. Inform. Appl. 46, 165–179 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Silva, P.V.: Fixed points of endomorphisms of virtually free groups. Pac. J. Math. 263(1), 207–240 (2013)

    Article  MATH  Google Scholar 

  21. Sims, C.C.: Computation with Finitely Presented Groups. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  22. Sykiotis, M.: Fixed points of symmetric endomorphisms of groups. Int. J. Algebra Comput. 12(5), 737–745 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge support from the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011. The first author also acknowledges the support of the FCT project SFRH/BPD/65428/2009.

Both authors are grateful to the anonymous referee by very good advice which contributed to improve the original version.

Author information

Authors and Affiliations

  1. Centro de Matemática, Faculdade de Ciências, Universidade do Porto, R. Campo Alegre 687, 4169-007, Porto, Portugal

    Emanuele Rodaro & Pedro V. Silva

Authors
  1. Emanuele Rodaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro V. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pedro V. Silva.

Additional information

Communicated by Jean-Eric Pin.

Dedicated to the memory of John M. Howie.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rodaro, E., Silva, P.V. Fixed points of endomorphisms of trace monoids. Semigroup Forum 89, 266–279 (2014). https://doi.org/10.1007/s00233-013-9553-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9553-0

Keywords

Search

Navigation