Meet Uniform Continuous Posets

DOI：10.3770/j.issn:2095-2651.2019.05.003

 作者 单位 毛徐新 南京航空航天大学理学院, 江苏 南京 210016 徐罗山 扬州大学数学科学学院, 江苏 扬州 225002

作为一致连续偏序集概念的推广, 本文利用一致完备偏序集上的一致Scott集, 引入了交一致连续偏序集概念, 并探讨了交一致连续偏序集的性质及刻画. 主要结果有: (1)一致完备偏序集$L$是交一致连续的当且仅当对任意$x\in L$及一致Scott集$U$, $\uparrow\!(U\cap \downarrow x)$是一致Scott集; (2) 一致完备偏序集$L$是交一致连续的当且仅当对任意$x\in L$及一致集$S$, 有$x\wedge \bigvee S=\bigvee \{x\wedge s\mid s\in S\}$. 特别地, 一个完备格$L$是交一致连续的当且仅当$L$是完备Heyting代数; (3) 一致完备偏序集$L$是交一致连续的当且仅当$L$的任意主理想是交一致连续的当且仅当$L$的任意闭区间是交一致连续的当且仅当$L$的任意主滤子是交一致连续的; (4) 一致完备偏序集$L$是交一致连续的若$L$添加一最大元$1$后的$L^1$是完备Heyting代数; (5) 交一致连续偏序集的有限乘积和一致连续投射像仍然是交一致连续的.

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 $\uparrow\!(U\cap \downarrow x)$ is a uniform Scott set for each $x\in L$ and each uniform Scott set $U$; (2) A uniform complete poset $L$ is meet uniform continuous iff for each $x\in L$ and each uniform subset $S$, one has $x\wedge \bigvee S=\bigvee \{x\wedge s\mid s\in S\}$. 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 $L^1$ obtained by adjoining a top element 1 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.