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.


15A09 Theory of matrix inversion and generalized inverses
20M99 Semigroups
Full Text: DOI arXiv


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