( <i>w,v</i> )-core–EP inverse and ( <i>w,v</i> )-pseudo-core inverse

The core and core–EP inverses are two recent generalized inverses that were first introduced and studied for complex matrices, and later for elements of rings with involution. Afterward, weighted versions (the w-core–EP inverse and the weighted w-core inverse with weight v) were introduced. Our purpose is to study all these generalized inverses in the context of monoids with involution, in a unified way. With this in mind, we introduce new, broader kinds of generalized inverses. In particular, we present the (w,v)-core–EP inverse and the (w,v)-pseudo-core inverse for elements of a monoid with involution. Necessary and sufficient conditions for the existence of these inverses are proved, and some representations of these new inverses are developed. Using these results, we obtain new properties of the weighted w-core inverse with weight v. We finally consider these inverses in the matrix setting. We notably prove that these inverses provide minimum norm or least square solutions to some matrix equations.