Archimedean property of an element
Definition
Let \(S\) be a semigroup and let \(x \in S\). It is said that \(x\) is Archimedean if for all \(y \in S\), there exists a positive integer \(k\) and \(z \in S\) such that \(kx = y + z\). If each element in \(S\) is Archimedean, it is said that \(S\) is Archimedean.
Examples
\(\circ\) Let \(A = \{a_1, \ldots, a_n\}\) be a set of symbols, let \(*\) be the concatenation operation and let us consider the semigroup \(S = \{ c_1c_2 \cdots c_k ~ | ~ c_i \in A, i \in \{1, 2, \ldots, k\} \}\). No element of \(S\) is Archimedean. Indeed, given \(x = a_{i_1}\cdots a_{i_m}\) with \(i_1, \ldots, i_m \in \{1,2,\ldots, n\}\), for \(y = a_j\) with \(j \ne i_1\), there no exist \(k \in \mathbb{N} \setminus \{0\}\) and \(z \in S\) such that \(kx = y * z\).
\(\circ\) The semigroup \((\mathbb{N}^*, +)\), where \(\mathbb{N}^* = \mathbb{N} \setminus \{0\}\) is Archimedean. Given \(x, y \in \mathbb{N}^*\), let us consider \(k \in \mathbb{N}^*\) such that \(kx > y\). then, if we define \(z = kx - y > 0\), it is obtained that \(kx = y + z\).
\(\circ\) Given a numerical semigroup \(S\), it holds that \(S^* = S \setminus \{0\}\) is Archimedean. Let \(F(S)\) be the Frobenius number of \(S\). Given \(x, y \in S^*\), there exists \(k \in \mathbb{N} \setminus \{0\}\) such that \(kx - y > F(S)\). Then, \(z := kx - y \in S^*\) and \(kx = y + z\).