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θ-continuity and D θ-completion of posets

Published online by Cambridge University Press:  27 February 2017

ZHONGXI ZHANG ,
QINGGUO LI  and
XIAODONG JIA
ZHONGXI ZHANG
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China Email: liqingguoli@aliyun.com School of Computer Science, University of Birmingham, Birmingham, B15 2TT, U.K. Email: zhangzhongxi89@gmail.com, xxj312@cs.bham.ac.uk
QINGGUO LI
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China Email: liqingguoli@aliyun.com
XIAODONG JIA
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China Email: liqingguoli@aliyun.com School of Computer Science, University of Birmingham, Birmingham, B15 2TT, U.K. Email: zhangzhongxi89@gmail.com, xxj312@cs.bham.ac.uk
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Abstract

We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are D θ-completion and Ds2 -completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its D θ-completion is a domain; (2) the D θ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the D θ-completion D θ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 4 , April 2018 , pp. 533 - 547
Copyright
Copyright © Cambridge University Press 2017 

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