# SFB 303 Discussion Paper No. B-351

**Author:** Jain, S. K., S. K. Mitra, and Hans
Joachim Werner

**Title:** Extensions of $\Gscr$-based Matrix Partial Orders

**Abstract:** We prove that a partial order $\preceq^{\Gscr}$
on matrix partial order $\preceq^{\Gscr}$ on $\R^ {m\times n}$ can always be
extended to a $\Gscr$-based matrix partialorder $\preceq^{Gscr^*}$ such that
$\Gscr^* ( A ) \ne\emtyset$ for all $A\in R^ ( {m\times n }$, thus answering
an open Question [6]. It is further shown that $\Gscr6*$ should be
semicmplete. And even if in a special situation this is possible and even if
$\card\,\Gscr(A)\le 1$ for each $A$, then this does not mean that there also
need be a semicomplete extension such that $\Gscr^*(A) is 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.

**Keywords:** $\Gscr$-based matrix partial order, star order,
minus

**JEL-Classification-Number:**: C69

**Creation-Date:** 1995

**URL:**
../1995/bonnsfb351.pdf
SFB 303 Homepage

17.02.1998, © Webmaster