The CPO-inverse and its partial orders

This paper introduces the core projection operator inverse and its characterizations. Using this concept, we define the P-core partial order and D-core partial order, along with their properties. The P-core order is proven to be of minus-type, while the D-core order differs as a non-minus-type. We investigate connections between these orders and related ones like minus, star, and sharp partial orders for P-core, and minus and diamond partial orders for D-core. Furthermore, we present the P-core matrix, which intriguingly commutes with its core projection operator inverse. This work enhances our understanding of these concepts and their relationships.