The complete w-core inverse in semigroups

Let S be a *-semigroup and let a, w ∈ S. The initial goal of this work is to consider commuting properties of a new class of generalized inverses, called the complete w-core inverse of a, extending the EP invertibility of generalized inverses. It is shown that a is completely w-core invertible if and only if a is w-core invertible and a^{\#}_{w} commutes with aw if and only if a is w-EP invertible. Moreover, the representations of complete w-core inverses and w-EP inverses are both presented in rings. Also, the connections between the complete w-core inverse and other generalized inverses are given. More criterions for the complete w-core inverse is derived by units and one-sided ideals in rings.