Meet Uniform Continuous Posets 
Received:September 24, 2018 Revised:May 22, 2019 
Key Word:
uniform set uniform Scott set complete Heyting algebra meet uniform continuous poset principal ideal uniform continuous projection

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

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Abstract: 
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. 
Citation: 
DOI:10.3770/j.issn:20952651.2019.05.003 
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