Torsion free semigroup
Definition
Let \(S\) be a semigroup. It is said that \(S\) is torsion free if for any positive integer \(k\) and \(a,b \in S\) such that \(ka = kb\), then \(a = b\).
It can be proven that a commutative monoid \(A\) is isomorphic to a numerical semigroup if, and only if, \(A\) is finitely generated, quasi-Archimedean, torsion free and with only one idempotent.
Examples
\(\circ\) Every subset of \(\mathbb{N}\) is torsion free, in particular, every numerical semigroup is torsion free.
\(\circ\) Let \(S_3\) the group of permutations with three elements. If we consider the permutations \(\sigma_1 = (1,2), \sigma_2 = (2,3)\), we have that \(2\sigma_1 = \sigma_1 \circ \sigma_1 = Id = \sigma_2 \circ \sigma_2 = 2\sigma_2\), but \(\sigma_1 \ne \sigma_2\). Then, \(S_3\) is not a torsion free semigroup.