Properties and characterizations of dual sharp orders

Abstract

In this paper, we extend the sharp order to dual ring. To be specific, we introduce D-sharp, C-sharp, G- and G-sharp binary relations of dual matrices, and discuss relevant properties of these dual binary relations by using dual group generalized inverse(DGGI for short), group dual generalized inverse(GDGI for short) and matrix decomposition. We conclude that D-sharp and G-sharp binary relations are partial orders. The dual binary relations are connected with the sharp order by using the upper triangular block matrix. Furthermore, we discuss relationships among dual binary relations mentioned above. When dual parts of dual matrix pairs are all zero matrices, dual binary relations(D-sharp, C-sharp, G- and G-sharp binary relations) are equivalent to the sharp order.