Weakly Iterated Block Products of Finite Monoids

  • Howard Straubing
  • Denis Thérien
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2286)

Abstract

The block product of monoids is a bilateral version of the better known wreath product. Unlike the wreath product, block product is not associative. All decomposition theorems based on iterated block products that have appeared until now have assumed right-to-left bracketing of the operands. We here study what happens when the bracketing is made left-to-right. This parenthesization is in general weaker than the traditional one. We show that weakly iterated block products of semilattices correspond exactly to the well-known variety DA of finite monoids: if groups are allowed as factors, the variety DA*G is obtained. These decomposition theorems allow new, simpler, proofs of recent results concerning the defining power of generalized first-order logic using two variables only.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Eilenberg, Automata, Languages and Machines, vol. B, Academic Press, New York, 1976.MATHGoogle Scholar
  2. 2.
    J. E. Pin, Varieties of Formal Languages, Plenum, London, 1986.MATHGoogle Scholar
  3. 3.
    J. Rhodes and B. Tilson, “The Kernel of Monoid Morphisms”, J. Pure and Applied Algebra 62 (1989) 227–268.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. P. Schützenberger, “On finite monoids having only trivial subgroups”, Information and Control 8 (1965) 190–194.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. P. Schützenberger, “A remark on finite transducers”, Information and Control 4 (1961), 185–196.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. P. Schützenberger, “Sur le Produit de Concatenation Non-ambigu”, Semigroup Forum 13 (1976), 47–76.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Stiffler, “Extensions of the Fundamental Theorem of Finite Semigroups”, Advances in Mathematics, 11 159–209 (1973).CrossRefMathSciNetGoogle Scholar
  8. 8.
    H. Straubing and D. Thérien, “Regular languages defined by generalized first-order formulas with a bounded number of bound variables”, Proc. 18th Symposium on Theoretical Aspects of Computer Science 551–562 (2001).Google Scholar
  9. 9.
    H. Straubing, Finite Automata, Formal Logic and Circuit Complexity, Birkhäuser, Boston, 1994.Google Scholar
  10. 10.
    D. Thérien, “Two-sided wreath products of categories”, J. Pure and Applied Algebra 74 (1991) 307–315.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    D. Thérien and T. Wilke, “Over Words, Two Variables are as Powerful as One Quantifier Alternation,” Proc. 30th ACM Symposium on the Theory of Computing 256–263 (1998).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Howard Straubing
    • 1
  • Denis Thérien
    • 2
  1. 1.Boston CollegeUSA
  2. 2.McGill UniversityCanada

Personalised recommendations