Green index in semigroups: generators, presentations, and automatic structures.
Let \(T\) be a subsemigroup of a semigroup \(S\). \(T\)-relative Green relations [\textit{A. D. Wallace}, Acta Math. Acad. Sci. Hung. 14, 137-148 (1963; Zbl 0122.26802)] are defined for \(u,v\in S\) by \(u\mathcal R^Tv\Leftrightarrow uT^1=vT^1\), \(u\mathcal L^Tv\Leftrightarrow T^1u=T^1v\), \(\mathcal H^T=\mathcal R^T\cap\mathcal L^T\). The Green index of \(T\) in \(S\) is (the number of \(T\)-relative \(\mathcal H^T\)-classes in \(S\setminus T)+1\), i.e. the number of strong orbits in \(S\setminus T\) under the actions of \(T\) on \(S\). For each \(T\)-relative \(\mathcal H^T\)-class \(H\) the stabilizer of \(H\) in \(T\) is \(\text{Stab}(H)=\{t\in T^1:Ht=H\}\) and the relative Schützenberger group of \(H\) is \(\text{Stab}(H)/\gamma\), \((x,y)\in\gamma\Leftrightarrow (\forall h\in H)\;(hx=hy)\). Here is proved a rewriting theorem which shows how generating sets for \(S\) may be used to obtain generating sets for \(T\) and the Schützenberger groups and vice versa and presented a method for constructing a presentation for \(S\) from presentations of \(T\) and the Schützenberger groups. This allows to show that passing to finite Green index subsemigroups or extensions preserves finite generation, solubility of the word problem, growth type, automaticity for subsemigroups, finite presentability for extensions and finite Malcev presentability for group-embeddable semigroups.
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