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

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.