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Green index and finiteness conditions for semigroups. (English) Zbl 1172.20045
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.

##### MSC:
 20M10 General structure theory for semigroups
Full Text:
##### References:
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