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Zbl 0658.46006
Du, Yihong
Total order minihedral cones. (Chinese. English summary)
[J] J. Syst. Sci. Math. Sci. 8, No.1, 19-24 (1988). ISSN 1000-0577

A cone P in a Banach space E is called total order minihedral, if, under the partial ordering introduced by P, every upper bounded total ordering set in E has a minimal upper bound. The main results of this paper are the following. \par Theorem 1. Regular cones are total order minihedral, but the converse is not true. \par Theorem 2. If Banach space E is weakly sequentially complete, and P is a cone in E, then the following statements are equivalent: \par i) P is normal, ii) P is total order minihedral, iii) P is regular, iv) P is fully regular. \par Theorem 3. Suppose P is a total order minihedral cone, If, in addition, P is minihedral, then P is strongly minihedral \par Theorem 4. There exist total order minihedral cones which are not minihedral; there exist minihedral cones which are not total order minihedral.

MSC 2000:: *46A40 Ordered topological linear spaces

Keywords: Regular cones are total order minihedral; strongly minihedral

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