Abstract
The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the Möbius functions of a monoid and its Rees quotient are also provided.
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Apostol, T.M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, Berlin (1976)
Arendt, B.D., Stuth, C.J.: On partial homomorphisms of semigroups. Pac. J. Math. 35(1), 7–9 (1970)
Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs in Theoretical Computer Science, vol. 12. Springer, Berlin (1988)
Bourbaki, N.: Elements of Mathematics—Algebra. Springer, Berlin (1989), Chaps. 1–3
Bourbaki, N.: Elements of Mathematics—General Topology. Springer, Berlin (2006), Chaps. 5–10
Bourbarki, N.: Elements of Mathematics—Commutative Algebra. Springer, Berlin (2006), Chaps. 1–4
Brown, T.C.: On locally finite semigroups. Ukr. Math. J. 20, 631–636 (1968)
Cartier, P., Foata, D.: Problèmes Combinatoires de Commutation et Réarrangements. Lecture Notes in Mathematics, vol. 85. Springer, Berlin (1969)
Clifford, A.H.: Partial homomorphic images of Brandt groupoids. Proc. Am. Math. Soc. 16(3), 538–544 (1965)
Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1961)
Cohn, P.M.: Further Algebra and Applications. Springer, Berlin (2003)
Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Doubilet, P., Rota, G.-C., Stanley, R.P.: The idea of generating function. In: Rota, G.-C. (ed.) Finite Operator Calculus, pp. 83–134. Academic Press, San Diego (1975)
Duchamp, G.H.E., Krob, D.: Partially commutative formal power series. In: Proceedings of the LITP Spring School on Theoretical Computer Science on Semantics of Systems of Concurrent Processes, pp. 256–276 (1990)
Duchamp, G.H.E., Krob, D.: Plactic-growth-like monoids. In: Ito, M., Jürgensen, H. (eds.) Words, Languages and Combinatorics II, Kyoto, Japan 1992, pp. 124–142 (1994)
Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, San Diego (1974)
Ésik, Z.: Iteration semirings. In: Proceedings of the 12th International Conference on Developments in Language Theory, Kyoto, Japan. Lecture Notes in Computer Science, vol. 5257, pp. 1–20. Springer, Berlin (2008)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Grillet, P.A.: Semigroups: An Introduction to the Structure Theory. Monographs and Textbooks in Pure and Applied Mathematics. Dekker, New York (1995)
Hivert, F., Novelli, J.-C., Thibon, J.-Y.: Un analogue du monoïde plaxique pour les arbres binaires de recherche. C. R. Acad. Sci. Paris Sr. I Math. 332, 577–580 (2002)
Krob, D., Thibon, J.-Y.: Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at q=0. J. Algorithms Comb. 6(4), 339–376 (1997)
Mc Lane, S.: Categories for the Working Mathematician, 2nd edn. Graduate Texts in Mathematics, vol. 5. Springer, Berlin (1998)
Lascoux, A., Schützenberger, M.P.: Le monoïde plaxique. In: Luca, De, A. (ed.) Non-commutative Structures in Algebra and Geometric Combinatorics, Quaderni de la ricerca scientifica, vol. 109, pp. 129–156. C.N.R., Rome (1981)
Novelli, J.-C.: On the hypoplactic monoid. Discrete Math. 217(1), 315–336 (2000)
Novikov, B.V.: 0-cohomology of semigroups. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 5, pp. 189–210. Elsevier, Amsterdam (2008)
Novikov, B.V., Polyakova, L.Y.: On 0-homology of categorical at zero semigroups. Cent. Eur. J. Math. 7(2), 165–175 (2009)
Okniński, J.: Semigroup Algebras. Pure and Applied Algebra: A Series of Monographs and Textbooks, vol. 138. Dekker, Berlin (1990)
Shevrin, L.N.: On locally finite semigroups. Sov. Math. Dokl. 6, 769–772 (1965)
Stanley, R.P.: Enumerative Combinatorics, vol. 1. University Press, Cambridge (1996)
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Communicated by Dominique Perrin.
The three authors wish to acknowledge support from Agence Nationale de la Recherche (Paris, France) under Program No. ANR-08-BLAN-0243-2. One of us, Laurent Poinsot, wishes to acknowledge support from the University Paris-Nord XIII under Project BQR CoVAMP.
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Poinsot, L., Duchamp, G.H.E. & Tollu, C. Möbius inversion formula for monoids with zero. Semigroup Forum 81, 446–460 (2010). https://doi.org/10.1007/s00233-010-9265-7
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DOI: https://doi.org/10.1007/s00233-010-9265-7