Commuting and EP-like properties of outer inverses in semigroups and rings

In this paper, we give necessary and sufficient conditions for the commuting property to hold for several important outer inverses such as Drazin’s [Formula: see text]-inverses, hybrid [Formula: see text]-inverses and annihilator [Formula: see text]-inverses by studying the two conditions [Formula: see text] and [Formula: see text]. Moreover, we further investigate Drazin’s left and right [Formula: see text]-EP properties and introduce two new EP-like properties associated with them.