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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 elementsand their-outer generalized inversesandthrough which the reverse order law results aboutandare obtained.

Common properties among various generalized inverses and constrained binary relations