The Weighted Core-EP Inverse and Its Associated Preorder

In this paper, we introduce a new preorder derived from the w-core-EP inverse in a *-ring. We characterize this generalized inverse by combining the w-core inverse with nilpotent elements. This characterization allows us to define a new binary relation on w-core-EP invertible elements, based on the w-core preorder. Using the Pierce matrix decomposition for two idempotents as a novel tool, we establish equivalent conditions for the forward and reverse order laws governing w-core-EP invertibility. Furthermore, we extend the *-DMP property to a broader class of elements within the framework of the w-core-EP preorder.