Almost symmetric good semigroups (Q6635327)

Let \(\mathbb{N}\) be the set of nonnegative integers, and let \(h \geq 2\) be an integer. As usual, \(\leq\) denotes the natural partial ordering on \(\mathbb{N}^h\): \(\alpha \leq \beta\) if \(\alpha_i \leq \beta_i\) for every \(i=1, \dots, h\). For \(\alpha, \beta \in \mathbb{N}^h\), the infimum of the set \(\{\alpha, \beta\}\) (with respect to \(\leq\)) is denoted by \(\alpha \wedge \beta\). Therefore, \(\alpha \wedge \beta = (\min(\alpha_1, \beta_1), \dots, \min(\alpha_h, \beta_h))\). A submonoid \(S \subseteq \mathbb{N}^h\) is called a good semigroup if it satisfies the following three properties: (G1) If \(\alpha, \beta \in S\), then \(\alpha \wedge \beta \in S\). (G2) If \(\alpha, \beta \in S, \alpha \neq \beta\), and \(\alpha_i = \beta_i\) for some \(i \in \{1, \dots, h\}\), then there exists \(\delta \in S\) such that \(\delta_i > \alpha_i = \beta_i\) and \(\delta_j \geq \min(\alpha_j, \beta_j)\) for all \(j \neq i\) (with equality if \(\alpha_j \neq \beta_j\)). (G3) There exists \(c \in \mathbb{N}^h\) such that \(c + \mathbb{N}^h \subseteq S \). Note that when \(h = 1\), properties (G1) and (G2) become trivial, and property (G3), for a submonoid \(S\) of \(\mathbb{N}\), implies that \(S\) is a numerical semigroup. Thus, in the case where \(h = 1\), we can think of numerical semigroups as good semigroups. So The class of good semigroups is a family of subsemigroups of \(\mathbb{N}^h\) that includes the value semigroups of rings associated with curve singularities and their blowups. This class facilitates a combinatorial study of the properties of these rings. In this paper, the authors provide a characterization of almost symmetric good subsemigroups of \(\mathbb{N}^h\), extending existing results from numerical semigroup theory and one-dimensional ring theory. They also apply these findings to derive new results concerning almost Gorenstein one-dimensional analytically unramified rings.