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.
Theorem 1. Regular cones are total order minihedral, but the converse is not true.
Theorem 2. If Banach space E is weakly sequentially complete, and P is a cone in E, then the following statements are equivalent:
i) P is normal, ii) P is total order minihedral, iii) P is regular, iv) P is fully regular.
Theorem 3. Suppose P is a total order minihedral cone, If, in addition, P is minihedral, then P is strongly minihedral
Theorem 4. There exist total order minihedral cones which are not minihedral; there exist minihedral cones which are not total order minihedral.