{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:35:53Z","timestamp":1787322953410,"version":"3.56.0"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1996,10]]},"abstract":"<jats:p>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 ) \\ne \\emptyset $ for all $A \\in {\\bf R}^{m \\times n} $, thus answering an open question [Mitra, Linear Algebra Appl., 148 (1991), pp. 237\u2013263]. 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.<\/jats:p>","DOI":"10.1137\/0617051","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"834-850","source":"Crossref","is-referenced-by-count":5,"title":["Extensions of $\\mathcal{G}$-Based Matrix Partial Orders"],"prefix":"10.1137","volume":"17","author":[{"given":"S. K.","family":"Jain","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S. K.","family":"Mitra","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. J.","family":"Werner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","volume-title":"Generalized inverses: theory and applications","author":"Ben-Israel Adi","year":"1974"},{"key":"R1","volume-title":"Generalized inverses: theory and applications","author":"Ben-Israel Adi","year":"1980"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1964.10480751"},{"key":"R3","unstructured":"J. Hartung, H. J. 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