Meet Uniform Continuous Posets  

作　　者：Xuxin MAO Luoshan XU 

机构地区：[1]College of Science, Nanjing University of Aeronautics and Astronautics, Jiangsu 210016, P. R. China [2]Department of Mathematics, Yangzhou University, Jiangsu 225002, P. R. China

出　　处：《Journal of Mathematical Research with Applications》2019年第5期459-468,共10页数学研究及应用（英文版）

基　　金：Supported by the National Natural Science Foundation of China(Grant Nos.11671008; 11101212);the Natural Science Foundation of Jiangsu Province(Grant No.BK20170483);the Fund of University Speciality Construction of Jiangsu Province(Grant No.PPZY2015B109)

摘　　要：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 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.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.

关 键 词：UNIFORM SET UNIFORM SCOTT SET complete Heyting algebra MEET UNIFORM CONTINUOUS POSET principal ideal UNIFORM CONTINUOUS projection 

分 类 号：O1[理学—数学]