Reidemeister-Schreier type rewriting for semigroups
Let \(T\) be a subsemigroup of a semigroup \(S\). The index of \(T\) in \(S\) is defined as \(1+\text{card}(S-T)\). The authors give a rewriting method that yields a presentation for \(T\) from a given presentation for \(S\). This general method is not constructive. However, when \(T\) is a two sided ideal of \(S\), the authors produce a rewriting procedure similar to the Reidemeister-Schreier rewriting theorem for groups. The methods of proof have similarities with those for groups. However, the presentation and the proof are rather different and technically more complicated than those for groups. For any semigroup \(S\), \(\text{rank}(S)\) is the least cardinality of a generating set for \(S\). Corollaries of the authors' presentations are: For any subsemigroup \(T\) of index \(i\) in a semigroup \(S\), \(\text{rank}(T)\leq i^2\text{rank}(S)\). If \(S\) is finitely presented and \(T\) is a two sided ideal of \(S\) of finite index, then \(T\) is finitely presented. The authors study also presentations for the minimal ideal of a semigroup \(S\) and its Schützenberger group. Suppose \(I\) is a 0-completely simple minimal 0-ideal of a finitely presented semigroup \(S\). If \(I\) has finitely many 0-minimal left ideals or finitely many 0-minimal right ideals, then its Schützenberger group is finitely presented. If \(I\) has finitely many 0-minimal left ideals and finitely many 0-minimal right ideals, then \(I\) is finitely presented.
scientific article; zbMATH DE number 784130