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Some results on C-varieties

Published online by Cambridge University Press:  15 March 2005

Jean-Éric Pin and
Howard Straubing
Jean-Éric Pin
Affiliation:
LIAFA, Université Paris VII and CNRS, Case 7014, 2 Place Jussieu, 75251 Paris Cedex 05, France; Jean-Eric.Pin@liafa.jussieu.fr
Howard Straubing
Affiliation:
Department of Computer Science, Boston College, Chestnut Hill, MA 02167, USA; straubin@cs.bc.edu
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Abstract

In an earlier paper, the second author generalized Eilenberg's variety theory by establishing a basic correspondence between certain classes of monoid morphisms and families of regular languages. We extend this theory in several directions. First, we prove a version of Reiterman's theorem concerning the definition of varieties by identities, and illustrate this result by describing the identities associated with languages of the form (a1a2...ak) +, where a1,...,ak are distinct letters. Next, we generalize the notions of Mal'cev product, positive varieties, and polynomial closure. Our results not only extend those already known, but permit a unified approach of different cases that previously required separate treatment.

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References

Barrington, D.A.M., Compton, K.J., Straubing, H. and Thérien, D., Regular languages in nc 1. J. Comput. Syst. Sci. 44 (1992) 478–499. CrossRef
M. Branco, Varieties of languages, in Semigroups, Algorithms, Automata and Languages, edited by G.M.S. Gomes, J.-É. Pin and P. Silva. World Scientific (2002) 91–132.
S. Eilenberg, Automata, languages, and machines. Vol. B. Academic Press, Harcourt Brace Jovanovich Publishers, New York, (1976). With two chapters (“Depth decomposition theorem” and “Complexity of semigroups and morphisms”) by Bret Tilson. Pure Appl. Math. 59.
Eilenberg, S. and Schützenberger, M.-P., On pseudovarieties. Adv. Math. 19 (1976) 413–418. CrossRef
Kunc, M., Equational description of pseudovarieties of homomorphisms. Theor. Inform. Appl. 37 (2003) 243–254. CrossRef
D. Perrin and J.-É. Pin, Infinite Words. Pure and Applied Mathematics 141 2004.
Pin, J.-É., A variety theorem without complementation. Russian Math. (Izvestija vuzov. Matematika) 39 (1995) 80–90.
J.-É. Pin, Syntactic semigroups, in Handbook of formal languages, edited by G. Rozenberg and A. Salomaa. Springer-Verlag 1 (1997) 679–746.
Pin, J.-É., Algebraic tools for the concatenation product. Theoret. Comput. Sci. 292 (2003) 317–342. CrossRef
Pin, J.-É., Straubing, H. and Thérien, D., Some results on the generalized star-height problem. Inform. Comput. 101 (1992) 219–250. CrossRef
Pin, J.-É. and Weil, P., Profinite semigroups, mal'cev products and identities. J. Algebra 182 (1996) 604–626. CrossRef
Pin, J.-É. and Weil, P., Polynomial closure and unambiguous product. Theory Comput. Syst. 30 (1997) 1–39. CrossRef
Pin, J.-É. and Weil, P., Semidirect products of ordered semigroups. Commun. Algebra 30 (2002) 149–169. CrossRef
Reiterman, J., The Birkhoff theorem for finite algebras. Algebra Universalis 14 (1982) 1–10. CrossRef
I. Simon, Hierarchies of Events with Dot-Depth One. Ph.D. Thesis, University of Waterloo, Waterloo, Ontario, Canada (1972).
Simon, I., Piecewise testable events, in Proc. 2nd GI Conf., edited by H. Brackage. Springer-Verlag, Berlin, Heidelberg, New York. Lect. Notes Comp. Sci. 33 (1975) 214–222. CrossRef
H. Straubing, Finite automata, formal logic, and circuit complexity. Birkhäuser Boston Inc., Boston, MA (1994).
H. Straubing, On logical descriptions of regular languages, in LATIN 2002. Springer, Berlin, Lect. Notes Comput. Sci. 2286 (2002) 528–538.