Abstract
This article is devoted to the study of monoids which can be endowed with a shuffle product with coefficients in a semiring. We show that, when the multiplicities do not belong to a ring with prime characteristic, such a monoid is a monoid of traces. When the characteristic is prime, we give a decomposition of the congruences ≡ (or relators R) such that A*/≡=(A; R) admits a shuffle product. This decomposition involves only addition of primitive elements to the successive quotients. To end with, we study the compatibility with Magnus transformation and examine the case of congruences which are homogeneous for some weight function. The existence of such a weight function is also showed for congruences of depth one.
Partially supported by the A.C. grant of MENRT.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Berstel and C. Reutenauer, Rational Series and Their Languages, ( EATCS Monographs on Theoretical Computer Science, Springer-Verlag Berlin, 1988 ).
N.Bourbaki, Éléments de mathématiques, Groupes et algèbres de Lie, Chap. 2 et 3 ( Hermann, Paris, 1972 ).
P.Cartier, D.Foata, Problèmes combinatoires de commutation et réarrangement, Lect. Not. In Math., n 85, 1969.
K.T.Chen, R.H.Fox, R.C.Lyndon, Free differential calculus IV- The quotient groups of the lower central series, Ann. Of Math, 1958.
V. Diekert and G. Rozenberg, The book of traces ( World Scientific, Singapour, 1995 ).
G. Duchamp, M. Flouret, E. Laugerotte, Operations over Automata with Multiplicities, in Automato implementation procedeeding WIA, J.M. Champarnaud, D.Maurel and D. Ziadi ecls,1660, 183–191, 1999.
G. Duchamp, D. Krob, Factorisations dans le monade partiellement commutatif libre, C.R. Acad. Sci. Paris, t. 312, série I (1991), 189–192.
G.Duchamp, D.Krob, Free partially commutative structures J. Algebra 156–2 (1993) 318–361.
G.Duchamp, D.Krob, Partially commutative Magnus transformation Int. J. of Alg. And comp., 3–1, 1993, 15–41.
G.Duchamp, J.G.Luque, Transitive Factorizations,Colloque FPSAC’99 Barcelone,1999.
M.Flouret, Contribution à l’algorithmique non commutative, Thèse de doctorat, Univerité de Rouen, 1999.
P. Gastin, Decidability of the Star problem in A* x {b}*, Information Processing Letters, 44, 65–71, 1992.
S. Gaubert, J.Mairesse, Medeling and analysis of timed Petri nets using heap of pieces, IEEE,Trans. Autom. Control, Vol 44, n4, 683–697, 1999.
S.C. Kleene, Representation of events in nerve nets and finite automata, Automata Studies, Princeton Univ. Press (1956), 3–42.
D. Krob and P.Lalonde, Partially commutative Lyndon words Lect. Notes in Comput. Sci. 665 (1993) 237–246.
P.Lalonde, Contribution à l’étude des empilements ( Thèse de doctorat, LACIM, 1991 ).
M.Lothaire, Combinatorics on words,Addison Wesley, 1983.
P. Ochsenschläger, Binomialkoeffitzenten und Shuffle-Zahlen, Technischer Bericht, Fachbereich Informatik, T. H. Darmstadt,1981.
W.Schmitt, Hopf algebras and identities in free partially commutative monoids, T.C.S. North Holland, 1990.
J.Y. Thibon, Intégrité des algèbres de séries formelles sur un alphabet partiellement commutatif,T.C.S (1985), North-Holland.
X.G.Viennot, Heaps of pieces I: Basic definitions and combinatorial lemmas In G. Labelle et al., editors, Proceeding Combinatoire énumeratice, Montréal Quebec 1985, nymber 1234 in Lectures notes in Mathematics, 321–350, BerlinHeidelberg-New York, 1986, Springer.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duchamp, G., Luque, JG. (2000). Congruences Compatible with the Shuffle Product. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds) Formal Power Series and Algebraic Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04166-6_38
Download citation
DOI: https://doi.org/10.1007/978-3-662-04166-6_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08662-5
Online ISBN: 978-3-662-04166-6
eBook Packages: Springer Book Archive