Partition of the complement of good semigroup ideals and Apéry sets

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

“…We recall here some general properties concerning good semigroups and complementary sets of a good ideals proved in Section 1 and Section 2 of the paper [18]. We refer to that paper for all the necessary proofs.…”

“…From the proof of [18,Proposition 1.7], it follows also that we can choose each G i such that ∆ E H (α) = ∅ for every H G i . Using the definition of complete infimum we can define a canonical partition of the complementary set A of a given good ideal E. Let E a good ideal of a good semigroup S and A = S \ E. Given α = (α 1 , α 2 , .…”

“…In this article we discuss several applications of the construction of the Apéry set of a good semigroup given in [18]. Good semigroups form a family of submonoids of N d defined axiomatically in [3], in order to study Noetherian analytically unramified one-dimensional semilocal reduced rings, e.g.…”

“…More recently, good semigroups in N d with d ≥ 2 have been studied, still in connection with the geometric and algebraic theory of curve singularities, but also with the purpose of extending pure combinatoric properties of numerical semigroups to this more general setting. 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].…”

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In this article we discuss some applications of the construction of the Apéry set of a good semigroup in N d given in [18]. In particular we study: the duality of a symmetric and almost symmetric good semigroup, the Apéry set of non-local good semigroups and the Apéry set of value semigroups of plane curves.

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