The common characterizations and various individual properties of different generalized inverses are established. Several equivalent conditions for the core-EP, weak group inverse, m-weak group inverse and weak core inverse are presented. The brand new explicit expressions for the operator binary relations defined by various generalized inverses are obtained. We derive properties and study the relationship between two different elements $A,B\in \mathcal{B}\mathcal{(}\mathcal{H}\mathcal{)}$ and their $(T,S)$-outer generalized inverses $A_{T,S}^{(2)}$ and $B_{T,S}^{(2)}$ through which the reverse order law results about $(AB{)}^{◯\mathrm{w}}=B^{◯\mathrm{w}}{A}^{◯\mathrm{w}}$ and $(AB{)}^{◯\mathrm{d}}=B^{◯\mathrm{d}}{A}^{◯\mathrm{d}}$ are obtained.

中文翻译：

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各种广义逆和约束二元关系之间的共同性质

建立了不同广义逆的共同特征和各种个体性质。给出了核心-EP、弱群逆、m-弱群逆和弱核逆的几个等价条件。得到了由各种广义逆定义的算子二元关系的全新显式。我们导出属性并研究两个不同元素之间的关系$A,乙\in \mathcal{乙}\mathcal{(}\mathcal{H}\mathcal{)}$和他们的$(吨,\mathrm{小号})$-外广义逆$A_{吨,\mathrm{小号}}^{(\mathrm{2个})}$和${乙}_{吨,\mathrm{小号}}^{(\mathrm{2个})}$逆序法则由此产生$(A乙{)}^{◯\mathrm{w}}={乙}^{◯\mathrm{w}}{A}^{◯\mathrm{w}}$和$(A乙{)}^{◯\mathrm{d}}={乙}^{◯\mathrm{d}}{A}^{◯\mathrm{d}}$获得。