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).
 Author Name Affiliation Xuxin MAO College of Science, Nanjing University of Aeronautics and Astronautics, Jiangsu 210016, P. R. China Luoshan XU Department of Mathematics, Yangzhou University, Jiangsu 225002, P. R. China
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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.