We provide existence criteria and characterizations for outer inverses in a semigroup belonging to the prescribed Green’s \({\mathscr {R}}\)-, \({\mathscr {L}}\)- and \({\mathscr {H}}\)-classes. These results generalize the well-known problem of finding outer inverses of a matrix over a field with the prescribed range or/and null space. Outer inverses belonging to the prescribed Green’s \({\mathscr {R}}\)- and \({\mathscr {L}}\)-classes also represent extensions of Drazin’s (b, c)-inverses and Mary’s inverses along an element. We provide an overview of other such extensions that have emerged recently and compare them with the extensions introduced in this paper.

我们提供属于规定格林的半群中的外逆元的存在准则和特征}}\) - 类。这些结果概括了在具有规定范围或/和零空间的域上查找矩阵的外逆矩阵的众所周知的问题。属于规定 Green 的\({\mathscr {R}}\) - 和\({\mathscr {L}}\)类的外逆也表示 Drazin 的 ( b ,  c)-逆元和玛丽沿元素的逆元。我们概述了最近出现的其他此类扩展，并将它们与本文中介绍的扩展进行比较。