In this paper, we prove that the dual core generalized inverse of a dual matrix \(\hat{A}\) is the unique solution of \(\hat{A}\hat{X}=\hat{A}\hat{A}^{\dagger }\) and \(\textrm{R}(\hat{X}){\subseteq }\textrm{R}(\hat{A})\) . We present some new properties of D-core order, which is provided by Sitha and Mosić recently. Furthermore, we introduce two new dual partial orders: C-core order and Dual-minus core order. These three dual partial orders are all Dual-minus-type partial orders. In addition, we study properties, characterizations, and the relationships between these partial orders and Dual-minus partial orders.
