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Möbius inversion formula for monoids with zero

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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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Authors and Affiliations

  1. LIPN–UMR 7030, CNRS-Université Paris 13, 93430, Villetaneuse, France

    Laurent Poinsot, Gérard H. E. Duchamp & Christophe Tollu

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  1. Laurent Poinsot
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  2. Gérard H. E. Duchamp
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  3. Christophe Tollu
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Correspondence to Laurent Poinsot.

Additional information

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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