Semigroup Forum

, Volume 84, Issue 3, pp 515–526 | Cite as

Finite derivation type for semilattices of semigroups



In this paper we investigate how the combinatorial property finite derivation type (FDT) is preserved in a semilattice of semigroups. We prove that if \(S= \mathcal{S}[Y,S_{\alpha}]\) is a semilattice of semigroups such that Y is finite and each Sα (αY) has FDT, then S has FDT. As a consequence we can show that a strong semilattice of semigroups \(\mathcal{S}[Y,S_{\alpha},\lambda_{\alpha,\beta}]\) has FDT if and only if Y is finite and every semigroup Sα (αY) has FDT.


Finite derivation type Semigroup Presentation Squier complex Semilattice of semigroups Strong semilattice of semigroups 


  1. 1.
    Araújo, I.M., Branco, M.J.J., Fernandes, V.H., Gomes, G.M.S., Ruškuc, N.: On generators and relations for unions of semigroups. Semigroup Forum 63(1), 49–62 (2001) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs. New Series, vol. 12. The Clarendon Press, Oxford University Press, New York (1995). Oxford Science Publications MATHGoogle Scholar
  3. 3.
    Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, New York (1970) Google Scholar
  4. 4.
    Malheiro, A.: Finite derivation type for Rees matrix semigroups. Theor. Comput. Sci. 355(3), 274–290 (2006) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Malheiro, A.: On trivializers and subsemigroups. In: Semigroups and Formal Languages, pp. 188–204. World Sci. Publ., Hackensack (2007) Google Scholar
  6. 6.
    Malheiro, A.: Finite derivation type for large ideals. Semigroup Forum 78(3), 450–485 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Otto, F.: Modular properties of monoids and string-rewriting systems. In: Nehaniv, C., Ito, M. (eds.) Algebraic Engineering, pp. 538–554. World Scientific, Singapore (1999) Google Scholar
  8. 8.
    Pride, S.J.: Geometric methods in combinatorial semigroup theory. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups, pp. 215–232. Kluwer Academic, Dordrecht (1995) Google Scholar
  9. 9.
    Pride, S.J., Wang, J.: Rewriting systems, finiteness conditions, and associated functions. In: Birget, J.C., Margolis, S., Meakin, J., Sapir, M. (eds.) Algorithmic Problems in Groups and Semigroups, pp. 195–216. Birkhäuser, Boston (2000) CrossRefGoogle Scholar
  10. 10.
    Pride, S.J., Wang, X.: Second order Dehn functions of groups and monoids. Int. J. Algebra Comput. 10, 425–456 (2000) MathSciNetMATHGoogle Scholar
  11. 11.
    Squier, C.C., Otto, F., Kobayashi, Y.: A finiteness condition for rewriting systems. Theor. Comput. Sci. 131, 271–294 (1994) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Wang, J.: Finite complete rewriting systems and finite derivation type for small extensions of monoids. J. Algebra 204, 493–503 (1998) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Wang, J.: Finite derivation type for semi-direct products of monoids. Theor. Comput. Sci. 191, 219–228 (1998) MATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Centro de ÁlgebraUniversidade de LisboaLisboaPortugal

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