Advertisement

Join Irreducible Pseudovarieties, Group Mapping, and Kovács-Newman Semigroups

  • John Rhodes
  • Benjamin Steinberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

We call a pseudovariety finite join irreducible if

\(V\leq V_{1}v V_{2}\Longrightarrow V \leq V_{1} {\rm or} V \leq V_{2}. \)

We present a large class of group mapping semigroups generating finite join irreducible pseudovarieties. We show that many naturally occurring pseudovarieties are finite join irreducible including: S, DS, CR, CS and \(\overline{H}\), where H is a group pseudovariety containing a non-nilpotent group.

Keywords

Normal Subgroup Maximal Subgroup Nilpotent Group Group Mapping Minimal Normal Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)zbMATHGoogle Scholar
  2. 2.
    Auinger, K., Steinberg, B.: The geometry of profinite graphs with applications to free groups and finite monoids. Trans. Amer. Math. Soc. 356, 805–851 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Auinger, K., Steinberg, B.: On power groups and embedding theorems for relatively free profinite monoids. In: Proc. Cambridge Philos. Soc. (to appear)Google Scholar
  4. 4.
    Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press, New York (1976)zbMATHGoogle Scholar
  5. 5.
    Krohn, K., Rhodes, J., Tilson, B.: Lectures on the algebraic theory of finite semigroups and finite-state machines. In: Arbib, M.A. (ed.) Chapters 1, 5-9 (Chapter 6 with Arbib, M. A.). The Algebraic Theory of Machines, Languages, and Semigroups, Academic Press, New York (1968)Google Scholar
  6. 6.
    Kovács, L.G., Newman, M.F.: Cross varieties of groups. Proc. Roy. Soc (London) A 292, 530–536 (1966)zbMATHCrossRefGoogle Scholar
  7. 7.
    Kovács, L.G., Newman, M.F.: Minimal verbal subgroups. Proc. Cambridge Phil. Soc. 62, 347–350 (1966)zbMATHCrossRefGoogle Scholar
  8. 8.
    Kovács, L.G., Newman, M.F.: On critical groups. J. Austral. Math. Soc. 6, 237–250 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Neumann, H.: Varieties of Groups. Springer, Heidelberg (1967)zbMATHGoogle Scholar
  10. 10.
    Margolis, S.W., Sapir, M., Weil, P.: Irreducibility of certain pseudovarieties. Comm. Algebra 26, 779–792 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rhodes, J., Steinberg, B.: The q-theory of finite semigroups (in preparation), http://www.mathstat.carleton.ca/~bsteinbg/qtheor

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • John Rhodes
    • 1
  • Benjamin Steinberg
    • 2
  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

Personalised recommendations