Möbius inversion formula for monoids with zero

Laurent Poinsot (1) , Gérard Henry Edmond Duchamp (1) , Christophe Tollu (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

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.

Mots clés: Fonction de Möbius; monoïde à zéro; monoïde localement fini; quotient de Rees; algèbre contractée

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Laurent Poinsot, Gérard Henry Edmond Duchamp, Christophe Tollu. Möbius inversion formula for monoids with zero. Semigroup Forum, 2010, 81 (3), pp.446-460. ⟨10.1007/s00233-010-9265-7⟩. ⟨hal-00435918v2⟩