The tree of good semigroups in $\mathbb{N}^2$ and a generalization of the Wilf conjecture

Abstract

In this work, we study good semigroups of N n introducing the definition of length and genus for these objects. We show how to count the local good semigroup with a fixed genus. Furthermore we study the relationships of these concepts with other ones previously defined in the case of good semigroups with two branches.

Partition of complement of good ideals and Apéry sets

“…For the value semigroup of an analytically irreducible ring, this definition has an algebraic interpretation which is pointed up in [5,Corollary 1.6]. Furthermore, these sets has been used in [15,Proposition 1.6] to produce a formula for the computation of the genus of a good semigroup. Intuitively, they can be seen as lines, planes or higher dimensional subsets of N d that are all contained in the semigroup and more particularly in a good ideal or in its complement.…”

“…Unfortunately, a consequence of this fact is that, in order to prove some property for a good semigroup, it is not in general possible to take advantage of the nature of value semigroups of rings. For this reason, good semigroups with two branches have also been recently studied as a natural generalization of numerical semigroups, using only semigroup techniques, without necessarily referring to the ring context (see [7], [14], [8], [15]). More precisely, working in the case of good subsemigroups of N 2 , in [7], the authors study the notion of Apéry set, which is in this case infinite, but they show that it has a natural partition in a finite number of sets, called levels, and the number of these sets is e = e 1 + e 2 where (e 1 , e 2 ) is the minimal nonzero element in S. In case S is a value semigroup of a ring, this number is equal to the multiplicity of the corresponding ring agreeing with the analogous well-known result in the numerical case.…”