## On generalized inverses and Green’s relations.(English)Zbl 1219.15007

The paper deals with generalized inverses on semigroups by means of Green’s relations.
The author introduces a new type of generalized inverse, the inverse along an element, that is based on Green’s relations $${\mathcal L}$$, $${\mathcal R}$$ and $${\mathcal H}$$ and the associated preorders. Given a semigroup $$S$$ and elements $$a,b \in S$$, Green’s relations are defined by:
1) $$a {\mathcal L} b$$ if and only if $$S^1a=S^1b$$;
2) $$a {\mathcal R} b$$ if and only if $$aS^1=bS^1$$;
3) $$a {\mathcal H} b$$ if and only if $$a {\mathcal L} b$$ and $$a {\mathcal R} b$$,
where $$S^1$$ denotes the monoid generated by $$S$$, and the preorder relations are:
4) $$a \leq_{\mathcal L} b$$ if and only if $$S^1a \subset S^1b$$;
5) $$a \leq_{\mathcal R} b$$ if and only if $$aS^1 \subset bS^1$$;
6) $$a \leq_{\mathcal H} b$$ if and only if $$a \leq_{\mathcal L} b$$ and $$a \leq_{\mathcal R} b$$.
In this paper, the author defines the following generalized inverse: Given $$a,d \in S$$, $$b \in S$$ is an inverse of $$a$$ along $$d$$ if it verifies $$bad=d=dab$$ and $$b \leq_{\mathcal H} d$$. If moreover the inverse $$b$$ of $$a$$ along $$d$$ verifies $$aba=a$$, it is said that $$b$$ is an inner inverse of $$a$$ along $$d$$.
The author derives the properties of this new generalized inverse and shows that the classical generalized inverses: the group inverse, the Drazin inverse and the Moore-Penrose inverse, belong to this class, and retrieve their properties.

### MSC:

 15A09 Theory of matrix inversion and generalized inverses 20M99 Semigroups
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### References:

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