Left and right hybrid (b,c)-core inverses in semigroups with involution

Abstract

Let [Formula: see text] be a unital ∗-semigroup and let [Formula: see text]. The goal of this paper is to introduce two new classes of generalized inverses, called the left hybrid [Formula: see text]-core inverse and the right hybrid [Formula: see text]-core inverse. An element [Formula: see text] is left hybrid [Formula: see text]-core invertible if there exists some [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text]. Such an [Formula: see text] is called a left hybrid [Formula: see text]-core inverse of [Formula: see text]. Several criteria and properties for the left hybrid [Formula: see text]-core inverse are given. Among these, it is shown that [Formula: see text] is left hybrid [Formula: see text]-core invertible if and only if [Formula: see text] is left hybrid [Formula: see text]-invertible and [Formula: see text] is [Formula: see text]-invertible if and only if [Formula: see text] is left hybrid [Formula: see text]-invertible. The right hybrid [Formula: see text]-core inverse of [Formula: see text] is defined by the existence of [Formula: see text] satisfying [Formula: see text], [Formula: see text] and [Formula: see text]. Dual results for the right hybrid [Formula: see text]-core inverse are also demonstrated.