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

Abstract

Let S be a numerical semigroup and let (Z, ≤S) be the (locally finite) poset induced by S on the set of integers Z defined by x ≤S y if and only if y − x ∈ S for all integers x and y. In this paper, we investigate the Möbius function associated to (Z, ≤S) when S is an arithmetic semigroup.

numericalsgps, a GAP package for numerical semigroups

“…Furthermore, they are extensively used each time new algorithms are implemented and tests need to be made. [ 8,103 ], [ 25,109 ], [ 35,57,125 ], [ 3,52 ], [ 15,170,178 ], [ 3,145 ], [ 21,68,153 ] ] 2.13. Contributions.…”

numericalsgps, a GAP package for numerical semigroups

“…Recall that associated to the numerical semigroup S, we can define the partial order on Z, a ≤ S b if b − a ∈ S. Thus (Z, ≤ S ) is a poset, and one can define the Möbius function associated to it. We implement the procedure presented in [15].…”

“…gap> l:=FactorizationsIntegerWRTList(100, [10,11,13,15] (1 − x)H S (x) is a polynomial, which we call the polynomial associated to S (see [36]). We provide functions to compute both the polynomial and Hilbert series of a numerical semigroup.…”

“…Also to determine a basis of the algebra K F (or K[F]) such that the values (or degrees) minimally generate the semigroup of values of this algebra (see [5]). gap> t:=Indeterminate(Rationals,"t");; gap> l:=[tˆ4,tˆ6+tˆ7,tˆ13]; [ tˆ4, tˆ7+tˆ6, tˆ13 ] gap> SemigroupOfValuesOfCurve_Local(l); <Numerical semigroup with 4 generators> gap> MinimalGeneratingSystem(last); [ 4,6,13,15 ] gap> SemigroupOfValuesOfCurve_Local(l,"basis"); [ tˆ4, tˆ7+tˆ6, tˆ13, tˆ15 ] 2.11. Affine semigroups.…”

Chomp on numerical semigroups

“…. , x n ∈ N}.The semigroup S induces on itself a poset structure (S, ≤ S ) whose partial order ≤ S is defined byx ≤ S y ⇐⇒ y − x ∈ S. This poset structure on S has been considered in [11,15] to study algebraic properties of its corresponding semigroup algebra, and in [5,6,7] to study its Möbius function. We observe that…”