The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture

Abstract

Good subsemigroups of $${\mathbb {N}}^d$$ N d have been introduced as the most natural generalization of numerical ones. Although their definition arises by taking into account the properties of value semigroups of analytically unramified rings (for instance the local rings of an algebraic curve), not all good semigroups can be obtained as value semigroups, implying that they can be studied as pure combinatorial objects. In this work, we are going to introduce the definition of length and genus for good semigroups in $${\mathbb {N}}^d$$ N d . For $$d=2$$ d = 2 , we show how to count all the local good semigroups with a fixed genus through the introduction of the tree of local good subsemigroups of $${\mathbb {N}}^2$$ N 2 , generalizing the analogous concept introduced in the numerical case. Furthermore, we study the relationships between these elements and others previously defined in the case of good semigroups with two branches, as the type and the embedding dimension. Finally, we show that an analogue of Wilf’s conjecture fails for good semigroups in $${\mathbb {N}}^2$$ N 2 .

Properties and applications of the Apéry set of good semigroups in $\mathbb{N}^d$

“…The concept of Apéry set, classical notion in the theory of numerical semigroups, has been extended to the "good" case, first in [5] for value semigroups of plane curves with two branches, then for arbitrary good semigroups in N 2 in [12], and for any good semigroup and any d in [18]. This notion has been a fundamental tool to generalize various features of the numerical setting, obtaining new characterization of classes of good semigroups, such as symmetric and almost symmetric, and studying important invariants, such as type, embedding dimension, genus [12], [13], [22], [23]. Unfortunately, for non-numerical good semigroups, the Apéry set is an infinite set, but it can be partitioned canonically in a finite number of subsets, called levels.…”

“…12, Theorem 9] and[13, Theorem 5.6] after proving the next general result that we state as Theorem 3.4.Example 3.2. Let us consider the good semigroup S ⊆ N 3 , having elements Using the procedure described in[23, Proposition 1.6], it is possible to see that the length and genus of the good semigroup are both equal to the half of the sum of the components of the conductor. This imply that S is a symmetric good semigroup (see[16, Theorem 2.3]).…”

Properties and applications of the Apéry set of good semigroups in $$\mathbb {N}^d$$

“…This notion has been a fundamental tool to generalize various features of the numerical setting, obtaining new characterization of classes of good semigroups, such as symmetric and almost symmetric, and studying important invariants, such as type, embedding dimension, genus [12,13,22,23].…”