Skip to main content
SpringerLink
Log in

Tameness of Pseudovariety Joins Involving \({\sf R}\)

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract.

In this paper, we establish several decidability results for pseudovariety joins of the form \({\sf V}\vee{\sf W}\), where \({\sf V}\) is a subpseudovariety of \({\sf J}\) or the pseudovariety \({\sf R}\). Here, \({\sf J}\) (resp. \({\sf R}\)) denotes the pseudovariety of all \({\cal J}\)-trivial (resp. \({\cal R}\)-trivial) semigroups. In particular, we show that the pseudovariety \({\sf V}\vee{\sf W}\) is (completely) κ-tame when \({\sf V}\) is a subpseudovariety of \({\sf J}\) with decidable κ-word problem and \({\sf W}\) is (completely) κ-tame. Moreover, if \({\sf W}\) is a κ-tame pseudovariety which satisfies the pseudoidentity x1 ⋯ x r yω+1ztω = x1 ⋯ x r yztω, then we prove that \({\sf R}\vee{\sf W}\) is also κ-tame. In particular the joins \({\sf R}\vee{\sf Ab}\), \({\sf R}\vee{\sf G}\), \({\sf R}\vee{\sf OCR}\), and \({\sf R}\vee{\sf CR}\) are decidable.

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

  • D Albert R Baldinger J Rhodes (1992) ArticleTitleUndecidability of the identity problem for finite semigroups. J Symbolic Logic 57 179–192

    Google Scholar 

  • J Almeida (1990) ArticleTitleImplicit operations on finite \({\cal J}\)-trivial semigroups and a conjecture of I. Simon. J Pure Appl Algebra 69 205–218 Occurrence Handle10.1016/0022-4049(91)90019-X

    Article  Google Scholar 

  • Almeida J (1995) Finite Semigroups and Universal Algebra. Singapore: World Scientific

  • J Almeida (1999) ArticleTitleHyperdecidable pseudovarieties and the calculation of semidirect products. Internat J Algebra Comput 9 241–261 Occurrence Handle10.1142/S0218196799000163

    Article  Google Scholar 

  • Almeida J (2002) Finite semigroups: an introduction to a unified theory of pseudovarieties. In: Gomes G, Pin J-E, Silva P (eds) Semigroups, Algorithms, Automata and Languages, pp 3–64. Singapore: World Scientific

  • J Almeida A Azevedo (1985) ArticleTitleThe join of the pseudovarieties of \({\cal R}\)-trivial and \({\cal L}\)-trivial monoids. J Pure Appl Algebra 60 129–137 Occurrence Handle10.1016/0022-4049(89)90125-4

    Article  Google Scholar 

  • J Almeida A Azevedo M Zeitoun (1999) ArticleTitlePseudovariety joins involving \({\cal J}\)-trivial semigroups and completely regular semigroups. Internat J Algebra Comput 9 99–112 Occurrence Handle10.1142/S0218196799000072

    Article  Google Scholar 

  • J Almeida M Delgado (1999) ArticleTitleSur certains systèmes d’équations avec contraintes dans un groupe libre. Portugal Math 56 409–417

    Google Scholar 

  • J Almeida M Delgado (2001) ArticleTitleSur certains systèmes d’équations avec contraintes dans un groupe libre–addenda. Portugal Math 58 379–387

    Google Scholar 

  • J Almeida M Delgado (2005) ArticleTitleTameness of the pseudovariety of abelian groups. Internat J Algebra Comput 15 327–338 Occurrence Handle10.1142/S0218196705002311 Occurrence HandleMR2142087

    Article  MathSciNet  Google Scholar 

  • J Almeida PV Silva (2001) ArticleTitleSC-hyperdecidability of R. Theoret Comput Sci 255 569–591 Occurrence Handle10.1016/S0304-3975(99)00329-1

    Article  Google Scholar 

  • J Almeida B Steinberg (2000) ArticleTitleOn the decidability of iterated semidirect products with applications to complexity. Proc London Math Soc 80 50–74 Occurrence Handle10.1112/S0024611500012144

    Article  Google Scholar 

  • Almeida J, Steinberg B (2000) Syntactic and global semigroup theory: a synthesis approach. In: Algorithmic Problems in Groups and Semigroups, pp 1–23. Boston: Birkhäuser

  • J Almeida PG Trotter (2001) ArticleTitleHyperdecidability of pseudovarieties of orthogroups. Glasgow Math J 43 67–83

    Google Scholar 

  • J Almeida PG Trotter (2001) ArticleTitleThe pseudoidentity problem and reducibility for completely regular semigroups. Bull Austral Math Soc 63 407–433

    Google Scholar 

  • J Almeida P Weil (1997) ArticleTitleFree profinite \({\cal R}\)-trivial monoids. Internat J Algebra Comput 7 625–671 Occurrence Handle10.1142/S0218196797000289

    Article  Google Scholar 

  • J Almeida M Zeitoun (1997) ArticleTitleThe pseudovariety J is hyperdecidable. RAIRO Inform Théor Appl 31 457–482

    Google Scholar 

  • Almeida J, Zeitoun M (2004) An automata-theoretical approach to the word problem for ω-terms over R. Tech Report CMUP 2004-38, Univ Porto (submitted)

  • Almeida J, Zeitoun M (2004) The equational theory of ω-terms for finite \({\cal R}\)-trivial semigroups. Proceedings of the Workshop Semigroups and Languages. Singapore: World Scientific

  • CJ Ash (1991) ArticleTitleInevitable graphs: a proof of the type II conjecture and some related decision procedures. Internat J Algebra Comput 1 127–146 Occurrence Handle10.1142/S0218196791000079

    Article  Google Scholar 

  • K Auinger B Steinberg (2003) ArticleTitleOn the extension problem for partial permutations. Proc Amer Math Soc 131 2693–2703 Occurrence Handle10.1090/S0002-9939-03-06860-6

    Article  Google Scholar 

  • Azevedo A (1990) The join of the pseudovariety J with permutative pseudovarieties. In: Almeida J, Bordalo G, Dwinger P (eds) Lattices, Semigroups and Universal Algebra, pp 1–11. New York: Plenum

  • JC Costa (2004) ArticleTitleReducibility of joins involving some locally trivial pseudovarieties. Comm Algebra 32 3517–3535 Occurrence Handle10.1081/AGB-120039628 Occurrence HandleMR2097476

    Article  MathSciNet  Google Scholar 

  • JC Costa ML Teixeira (2004) ArticleTitleTameness of the pseudovariety LSl. Internat J Algebra Comput 14 627–654 Occurrence Handle10.1142/S0218196704001955

    Article  Google Scholar 

  • Th Coulbois A Khélif (1999) ArticleTitleEquations in free groups are not finitely approximable. Proc Amer Math Soc 127 963–965 Occurrence Handle10.1090/S0002-9939-99-04747-4

    Article  Google Scholar 

  • B Herwig D Lascar (2000) ArticleTitleExtending partial automorphisms and the profinite topology on free groups. Trans Amer Math Soc 352 1985–2021 Occurrence Handle10.1090/S0002-9947-99-02374-0

    Article  Google Scholar 

  • Kunen K (1980) Set Theory. Amsterdam: North-Holland

  • J-E Pin P Weil (1996) ArticleTitleProfinite semigroups, Mal’cev products and identities. J Algebra 182 604–626 Occurrence Handle10.1006/jabr.1996.0192

    Article  Google Scholar 

  • J Rhodes B Steinberg (2002) ArticleTitleProfinite semigroups, varieties, expansions and the structure of relatively free profinite semigroups. Internat J Algebra Comput 11 627–672 Occurrence Handle10.1142/S0218196701000784

    Article  Google Scholar 

  • Steinberg B (1998) On pointlike sets and joins of pseudovarieties. Internat J Algebra Comput 8: 203–234 (with an addendum by the author)

    Google Scholar 

  • B Steinberg (2001) ArticleTitleOn algorithmic problems for joins of pseudovarieties. Semigroup Forum 62 1–40 Occurrence HandleMR2107191

    MathSciNet  Google Scholar 

  • M Zeitoun (1994) ArticleTitleOn the decidability of the membership problem of the pseudovariety J∨B. Internat J Algebra Comput 4 47–64

    Google Scholar 

  • Zeitoun M (1996) On the join of two pseudovarieties. In: Almeida J, Gomes GMS, Silva PV (eds) Semigroups, Automata and Languages, pp 281–288. Singapore: World Scientific

Download references

Author information

Authors and Affiliations

  1. Universidade do Porto, Portugal

    Jorge Almeida

  2. Universidade do Minho, Braga, Portugal

    José Carlos Costa

  3. Université Paris 7 and CNRS, France

    Marc Zeitoun

Authors
  1. Jorge Almeida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Carlos Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marc Zeitoun
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partial support by FCT, through the Centro de Matemática da Universidade do Porto, is also gratefully acknowledged.

Partial support by FCT, through the Centro de Matemática da Universidade do Minho, is also gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Almeida, J., Carlos Costa, J. & Zeitoun, M. Tameness of Pseudovariety Joins Involving \({\sf R}\). Mh Math 146, 89–111 (2005). https://doi.org/10.1007/s00605-005-0324-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-005-0324-1

Search

Navigation