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偏序集上一致连续性的等价刻画与性质
引用本文:毛徐新,徐罗山.偏序集上一致连续性的等价刻画与性质[J].高校应用数学学报(A辑),2019,34(1):121-126.
作者姓名:毛徐新  徐罗山
作者单位:南京航空航天大学 理学院,江苏南京,210016;扬州大学 数学科学学院,江苏扬州,225002
基金项目:国家自然科学基金;国家自然科学基金;国家自然科学基金;江苏省自然科学基金;江苏高校品牌专业建设工程
摘    要: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.

关 键 词:一致集  一致完备  一致连续偏序集  S-超连续偏序集  SCOTT  S-集  主理想

Properties and characterizations of uniform continuous posets
MAO Xu-xin,XU Luo-shan.Properties and characterizations of uniform continuous posets[J].Applied Mathematics A Journal of Chinese Universities,2019,34(1):121-126.
Authors:MAO Xu-xin  XU Luo-shan
Institution:(College of Sci., Nanjing Univ. of Aeronautics and Astronautics, Nanjing 210016, China;School of Math. Sci., Yangzhou Univ., Yangzhou 225002, China)
Abstract: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.
Keywords:uniform set  uniform complete  uniform continuous poset  supercontinuous poset  Scott S-set  principal ideal
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