EXTENSIONS OF G-BASED MATRIX PARTIAL ORDERS

We prove that a partial order less than or equal to(G) on R(mxn) can a lways be extended to a G-based matrix partial order less than or equal to(G) such that G*(A) not equal 0 for all A is an element of R(mxn), thus answering an open question [Mitra, Linear Algebra Appl., 148 (19 91), pp. 237-263]. It is further shown that this result does not in ge neral remain true if besides G, we also insist that G be semicomplete . And even if in a special situation this is possible and if card G(A) less than or equal to 1 for each A, this does not mean that;there als o need be a semicomplete extension such that G(A) is a singleton for all A. In addition, some other interesting results on matrix partial o rders are given. For instance, a useful characterisation for a semicom plete map to induce a partial order on the set of square matrices is d erived.