交C-连续偏序集

Meet C-continuous posets

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.

The concept of terizations of meet C-continuity, meet C-continuity for posets is as well as relationships of meet introduced. Properties and charac- C-continuity with C-continuity and QC-continuity are given. Main results are： （1） A lattice which is also meet C-continuous must be distributive; （2） A bounded complete poset （bc-poset, for short） L is meet C-continuous iff ∨x ∈ L and every none-empty Scott closed set S for which ∨S exists, one has x ∧∨S = ∨{x ∧s ： s ∈S}; （3） A complete lattice is a complete Heyting algebra iff it is meet continuous and meet C-continuous; （4） A bounded complete poser is C-continuous iff it is meet C-continuous and QC-continuous; （5） Some counterexamples are constructed to show that a distributive complete lattice needn＇t be a meet C-continuous lattice and a meet C-continuous lattice needn＇t be a meet continuous lattice.