全序极小锥

本文引进全序极小锥的概念,讨论了全序极小锥与正则锥、正规锥、极小锥及强极小锥的关系,改进了1]中的几个结果和11]的主要定理。按照1]中定义,Banach 空间 E 中锥 P 称为强极小的,如在 P 诱导的半序下,E 中任何按序有上界的子集都有最小上界;P 称为极小的,如 E 中任二元 x,y 都有最小上界;P称为正规的,如(?)N>0,使得θ≤x≤y时,‖x‖≤N‖y‖;P 正规(?)(?)δ>0,使得 x,y∈P,‖x‖=‖y‖=1时,‖x+y‖≥δ(?)E 中任何序区间x,y]都有界(?)x_n≤z_n≤y_n,且 x_n→z,y_n→z 时必有 z_n→z(参看3]第三章);P 称为正则的,如 E 中任何单调递增且有上界的序列都是收敛的,即 x_1≤x_2≤…≤x_n≤…≤x_0,则

TOTAL ORDER MINIHEDRAL CONES

A cone P in a Banach space E is called total order minihedral,if,under the partial or-dering introduced by P,every upper bounded total ordering set in E has a minimal upperbound.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 sequence complete,and P is a cone in E,thenthe 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 minihedr-al,then P is strongly minihedralTheorem 4.There exist total order minihedral cones which are not minihedral;thereexist minihedral cones which are not total order minihedral.