Green index and finiteness conditions for semigroups. (Q958525)

Given a semigroup \(S\) and a subsemigroup \(T\), the Rees index of \(T\) in \(S\) is defined to be the cardinality of the complement \(S\setminus T\). In this article, in Section 2, the authors give basic properties of Green index. Then in Section 3 and Section 4, they prove the following main results of their paper: Let \(S\) be a semigroup and let \(T\) be a subsemigroup of \(S\) with finite Green index. Let \(\Gamma_i\) (\(i\in I\)) be the Schützenberger groups of the \(T\)-relative \(H\)-classes of the complement \(S\setminus T\). Then the following hold: (I) \(S\) is locally finite if and only if \(T\) is locally finite, in which case every group \(\Gamma_i\) is locally finite; (II) \(S\) is periodic if and only if \(T\) is periodic, in which case every group \(\Gamma_i\) is periodic; (III) \(S\) has finitely many right ideals if and only if \(T\) has finitely many right ideals (and the dual result for left ideals); (IV) \(S\) is residually finite if and only if \(T\) and \(\Gamma_i \) are all residually finite. In Section 5, the authors define the relationship between Green index and syntactic index. Finally, they give some applications of their results.