Extensions of $\mathcal{G}$-Based Matrix Partial Orders
S. K. Jain, Sujit Kumar Mitra, Hans‐Joachim Werner · SIAM Journal on Matrix Analysis and Applications · 1996
We prove that a partial order $\preceq^{\mathcal{G} } $ on ${\bf R}^{m \times n}$ can always be extended to a $\mathcal{G}$-based matrix partial order $ \preceq^{\mathcal{G}^* } $ such that $\mathcal{G}^* ( A ) e \emptyset $ for all $A \in {\bf R}^{m \times n} $, thus answering an open question [Mitra, Linear Algebra Appl., 148 (1991), pp. 237–263]. It is further shown that this result does not in general remain true if besides $\mathcal{G}$, we also insist that $\mathcal{G}^* $ be semicomplete. And even if in a special situation this is possible and if card $\mathcal{G}( A ) \leq 1$ for each A, this does not mean that there also need be a semicomplete extension such that $\mathcal{G}^* ( A )$ is a singleton for all A. In addition, some other interesting results on matrix partial orders are given. For instance, a useful characterization for a semicomplete map to induce a partial order on the set of square matrices is derived.