Skip to main content
SpringerLink
Log in

Decidability Versus Undecidability of the Word Problem in Amalgams of Inverse Semigroups

  • Conference paper
  • First Online:
Semigroups, Algebras and Operator Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 142))

Abstract

This paper is a survey of some recent results on the word problem for amalgams of inverse semigroups. Some decidability results for special types of amalgams are summarized pointing out where and how the conditions posed on amalgams are used to guarantee the decidability of the word problem. Then a recent result on undecidability is shortly illustrated to show how small is the room between decidability and undecidability of the word problem in amalgams of inverse semigroups.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Bennett, C.: Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973)

    Article  MATH  Google Scholar 

  2. Bennett, P.: Amalgamated free product of inverse semigroups. J. Algebra 198, 499–537 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bennett, P.: On the structure of inverse semigroup amalgams. Int. J. Algebra Comput. 7(5), 577–604 (1997)

    Article  Google Scholar 

  4. Cherubini, A., Mazzucchelli, M.: On the decidability of the word problem for amalgamated free products of inverse semigroups. Semigroup Forum 76(2), 309–329 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cherubini, A., Rodaro, E.: Amalgams vs Yamamura’s HNN-extensions of inverse semigroups. Algebra Colloq. 18(04), 647–657 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cherubini, A., Jajcayová, T., Rodaro, E.: Maximal subgroups of amalgams of finite inverse semigroups. Semigroup Forum 9(2), 401–424 (2015)

    Google Scholar 

  7. Cherubini, A., Meakin, J., Piochi, B.: Amalgams of free inverse semigroups. Semigroup Forum 54, 199–220 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cherubini, A., Meakin, J., Piochi, B.: Amalgams of finite inverse semigroups. J. Algebra 285, 706–725 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cherubini, A., Nuccio, C., Rodaro, E.: Multilinear equations in amalgams of finite inverse semigroups. Int. J. Algebra Comput. 21(01n02), 35–59 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cherubini, A., Nuccio, C., Rodaro, E.: Amalgams of finite inverse semigroups and deterministic context-free languages. Semigroup Forum 85(1), 129–146 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Haataja, S., Margolis, S., Meakin, J.: Bass-Serre theory for groupoids and the structure of full regular semigroup amalgams. J. Algebra 183, 38–54 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hall, T.E.: Finite inverse semigroups and amalgamation. In: Goberstein, S.M., Higgins, P.M. (eds.) Semigroups and Their Applications, pp. 51–56. Reidel, Dordrecht (1987)

    Chapter  Google Scholar 

  13. Jajcayová, T.: HNN-extensions of inverse semigroups. Ph.D. thesis at University of Nebraska-Lincoln Department of Mathematics and Statistics (1997)

    Google Scholar 

  14. Janin, D.: Toward a higher-dimensional string string theory for the modeling of computerized systems. Technical report RR-1477-13, LaBRI, IPB, Université de Bordeaux (2013)

    Google Scholar 

  15. Jones, P.R., Margolis, S.W., Meakin, J.C., Stephen, J.B.: Free products of inverse semigroups. Glasg. Math. J. 33, 373–387 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kellendonk, J.: The local structure of tiling and their integer group of coinvariance. Commun. Math. Phys. 187, 115–157 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kellendonk, J.: Topological equivalence of tilings. J. Math. Phys. 38, 1823–1842 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kimura, N.: On semigroups. Ph.D. thesis at Tulane University of Louisiana (1957)

    Google Scholar 

  19. Lawson, M.V.: Inverse Semigroups. The Theory of Partial Symmetries. World Scientific, River Edge (1998)

    Book  MATH  Google Scholar 

  20. Margolis, S., Meakin, J., Sapir, M.: Algorithmic problems in groups, semigroups and inverse semigroups. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups, pp. 147–214 (1995)

    Google Scholar 

  21. Meakin, J.: Inverse semigroups: some open questions (2012)

    Google Scholar 

  22. Morita, K.: Universality of a reversible two-counter machine. Theor. Comput. Sci. 168, 303–320 (1996)

    Article  MATH  Google Scholar 

  23. Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Matematicheskogo Instituta imeni VA Steklova 3(29), 44–143 (1955)

    Google Scholar 

  24. Paterson, A.L.T.: Grupoids, Inverse Semigroups, and Their Operator Algebras. Birkhauser Boston, Boston (1999)

    Book  Google Scholar 

  25. Petrich, M.: Inverse Semigroups. Wiley, New York (1984)

    MATH  Google Scholar 

  26. Rodaro, E.: Bicyclic subsemigroups in amalgams of finite inverse semigroups. Int. J. Algebra Comput. 20(1), 89–113 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rodaro, E., Cherubini, A.: Decidability of the word problem in Yamamura’s HNN-extensions of finite inverse semigroups. Semigroup Forum 77(2), 163–186 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rodaro, E., Silva, P.V.: Amalgams of inverse semigroups and reversible two-counter machines. J. Pure Appl. Algebra 217(4), 585–597 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sapir, M.V.: Algorithmic problems for amalgams of finite semigroups. J. Algebra 229(2), 514–531 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Steinberg, B.: A topological approach to inverse and regular semigroups. Pac. J. Math. 208(2), 367–396 (2003)

    Article  MATH  Google Scholar 

  31. Stephen, J.B.: Presentation of inverse monoids. J. Pure Appl. Algebra 198, 81–112 (1990)

    Article  MathSciNet  Google Scholar 

  32. Stephen, J.B.: Amalgamated free products of inverse semigroups. J. Algebra 208, 339–424 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The first author is grateful to the organizers of ICSAOT 2014 for inviting her to present these results and acknowledges support from PRIN project 2011 “Automi e Linguaggi Formali: Aspetti Matematici e Applicativi.” The last author acknowledges 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/UI0144/2013 as well as support from the FCT project SFRH/BPD/65428/2009.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy

    Alessandra Cherubini

  2. Centro de Matemática, Universidade do Porto, Rua Do Campo Alegre, 687, 4169-007, Porto, Portugal

    Emanuele Rodaro

Authors
  1. Alessandra Cherubini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emanuele Rodaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandra Cherubini .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology (CUSAT), Kochi, Kerala, India

    P G Romeo

  2. Department of Mathematics, University of Nebraska, Lincoln, Nebraska, USA

    John. C Meakin

  3. Department of Mathematics, University of Kerala, Thiruvananthapuram, Kerala, India

    A R Rajan

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Cherubini, A., Rodaro, E. (2015). Decidability Versus Undecidability of the Word Problem in Amalgams of Inverse Semigroups. In: Romeo, P., Meakin, J., Rajan, A. (eds) Semigroups, Algebras and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 142. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2488-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation