Core-EP decomposition and related relations revisited

We study relations that are induced by the core-EP decomposition. We revisit the core-EP preorder and extend the concepts of the core-minus, the c-minus, and the c-star partial orders from the set of all $n\times n$ complex matrices to the set of all core-EP invertible elements in a $\ast$-ring. Several characterizations of these relations are presented and thus some known results are generalized.