The complete $w$-core inverse in semigroups

Abstract

Let $S$ be a $\ast$-semigroup and let $a, w\in S$.
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 $a_{w}^{\tiny{\textcircled{\tiny\#}}}$ 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.