测度拓扑和连续偏序集的刻画

研究偏序集上的测度拓扑以及与其它内蕴拓扑间的关系,利用测度拓扑刻画了偏序集的连续性.构造了反例说明存在完全分配格,其上的测度拓扑不是连续格从而不是局部紧拓扑.     

偏序集上一致连续性的等价刻画与性质

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.     

偏序集上的测度拓扑与稠拓扑

In this paper,the concepts of the essential topology and the density topology of dcpos are generalized to the setting of general posets.Basic properties of the essential topology and relations with other intrinsic topologies are explored.Comparisons between the density topology and the measurement topology are made.Via the essential topology,the density topology and the measurement topology,we obtain properties and characterizations of bases of continuous posets.We also provide some new conditions for a continuous poset to be an algebraic poset.     

交一致连续偏序集

In this paper, as a generalization of uniform continuous posets, the concept of meet uniform continuous posets via uniform Scott sets is introduced. Properties and characterizations of meet uniform continuous posets are presented. The main results are:(1) A uniform complete poset L is meet uniform continuous iff ↑(U ∩↓ x) is a uniform Scott set for each x ∈ L and each uniform Scott set U;(2) A uniform complete poset L is meet uniform continuous iff for each∨∨x∈ L and each uniform subset S, one has x ∧S ={x ∧ s | s ∈ S}. In particular, a complete lattice L is meet uniform continuous iff L is a complete Heyting algebra;(3) A uniform complete poset is meet uniform continuous iff every principal ideal is meet uniform continuous iff all closed intervals are meet uniform continuous iff all principal filters are meet uniform continuous;(4) A uniform complete poset L is meet uniform continuous if L1 obtained by adjoining a top element1 to L is a complete Heyting algebra;(5) Finite products and images of uniform continuous projections of meet uniform continuous posets are still meet uniform continuous.     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

偏序集的φ-(交)连续性及其主理想刻画

证明φ-完备偏序集是（强）P连续的当且仅当该偏序集的任一主理想是（强）φ-连续的。在φ-完备偏序集中利用φ-S集族生成f-Scott拓扑，并由此引入φ-交连续偏序集概念。证明φ-完备偏序集是P交连续的当且仅当该偏序集的任一主理想是φ-交连续的。