Semigroup ring
Definition
Let \(S\) be a numerical semigroup, \(t\) be an indeterminant and \(\mathbb{K}\) be a field. It is defined the semigroup ring of \(S\) over \(\mathbb{K}\) as \(\mathbb{K}[t^s ~ | ~ s \in S] = \bigoplus_{s \in S} \mathbb{K}t^{s}\) and is denoted by \(\mathbb{K}[S]\). Note that \(\mathbb{K}[S]\) is a subring of the polynomial ring \(\mathbb{K}[t]\).
Examples
\(\circ\) Let us consider \(b \in \mathbb{N}\) an odd natural arbitrary but fixed and \(S = \langle 2, b \rangle\). It is easy to see that \(S = \{0, 2, 4, \ldots, b-1, \rightarrow \}\). Then \[ \mathbb{K}[S] = \{a_0 + a_1t^{s_1} + \cdots + a_nt^{s_n} ~ | ~ a_0, \ldots, a_n \in \mathbb{K}, ~~ s_0, s_1, \ldots, s_n \in S\}. \] In this case, \(s_i\) is even or greater or equal to \(b\) for all \(i \in \{1, 2, \ldots, n\}\).
References
https://gap-packages.github.io/ numericalsgps.