On the Möbius function of the locally finite poset associated with a numerical semigroup

Jonathan Chappelon (1) , Jorge Luis Ramírez Alfonsín (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

Let $S$ be a numerical semigroup and let $(\mathbb{Z},\leq_S)$ be the (locally finite) poset induced by $S$ on the set of integers $\mathbb{Z}$ defined by $x \leq_S y$ if and only if $y-x\in S$ for all integers $x$ and $y$. In this paper, we investigate the {\it M\"{o}bius function} associated to $(\mathbb{Z},\leq_S)$ when $S$ is an {\it arithmetic} semigroup.

Mots clés: Möbius function; poset; numerical semigroup; arithmetic semigroup

Domaines: Combinatoire [math.CO]

Jonathan Chappelon, Jorge Luis Ramírez Alfonsín. On the Möbius function of the locally finite poset associated with a numerical semigroup. Semigroup Forum, 2013, 87 (2), pp.313-330. ⟨10.1007/s00233-012-9458-3⟩. ⟨hal-00730229v2⟩