偏序集上一致连续性的等价刻画与性质

Properties and characterizations of uniform continuous posets

将一致小于关系移植到一般偏序集上,同时引入了上界小于关系,定义了偏序集的一致连续性和上界连续性.给出了一致连续偏序集的等价刻画,探讨了一致连续偏序集所具有的性质.主要结果有:(1)证明了偏序集上的一致连续性,上界连续性与s-超连续性均等价;(2)在交半格条件下,偏序集的一致连续性等价于它的每一主理想一致连续;(3)在并半格条件下,偏序集的一致连续性蕴含连续性,反之不成立;(4)一致完备的一致连续偏序集均是连续bc-dcpo,且每个主理想均为完全分配格;(5)在一致完备的条件下,一致连续性对主滤子,对闭区间,对Scott S-集以及对一致连续投射像均是可遗传的.文中也构造了若干实用的反例.

In terms of the uniform way-below relations and the upper bound way-below relations defined on posets, the concepts of uniform continuous posets and upper bound continuous posets are introduced. Some characterizations and properties of uniform continuous posets are given. Main results are:(1) A poset is uniform continuous iff it is supercontinuous iff it is upper bound continuous;(2) A semilattice is uniform continuous iff every principal ideal is uniform continuous;(3) Every uniform continuous sup-semilattice is continuous, but the converse does not hold;(4) Every uniform complete poset which is uniform continuous is a continuous bc-dcpo with every principal ideal being a completely distributive complete lattice;(5) The uniform continuity on a uniform complete poset is hereditary for all principal ideals, principal filters, closed intervals, Scott-S sets and the images of uniform continuous projections. Some counterexamples are also constructed.