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θ-continuity and Dθ-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 Dθ-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 Dθ-completion is a domain; (2) the Dθ-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 Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

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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References

Abramsky, S. and Jung, A. (1994). Domain theory. In: Abramsky, S. et al. (eds.) Handbook of Logic in Computer Science, vol. 3, Clarendon Press, 1–168.Google Scholar
Erné, M. (1981). Scott convergence and Scott topology in partially ordered sets II. In: Banaschewski, B. and Hoffmann, R.-E. (eds.) Continuous Lattices, Proceedings, Bremen 1979. Lecture Notes on Mathematics, vol. 871, Springer Verlag, Berlin, 61–96.Google Scholar
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M. and Scott, D.S. (1980). A Compendium of Continuous Lattices, Springer, Berlin.Google Scholar
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M. and Scott, D.S. (2003). Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press.Google Scholar
Gierz, G., Lawson, J.D. and Stralka, A. (1983). Quasicontinuous posets. Houston Journal of Mathematics 9 (2) 191–208.Google Scholar
Heckmann, R. (1992). An upper power domain construction in terms of strongly compact sets. In: Brookes, S., Main, M., Melton, A., Mislove, M. and Schmidt, D. (eds.) Mathematical Foundations of Programming Semantics, Pittsburgh 1991. Lecture Notes in Computer Science, vol. 598, Springer, Berlin, 272–293.Google Scholar
Huang, M., Li, Q. and Li, J. (2009). Generalized continuous posets and a new cartesian closed category. Applied Categorical Structures 17 (1) 29–42.Google Scholar
Keimel, K. and Lawson, J.D. (2009). D-completions and the d-topology. Annals of Pure and Applied Logic 159 (3) 292–306.CrossRefGoogle Scholar
Keimel, K. and Lawson, J.D. (2012). Extending algebraic operations to D-completions. Theoretical Computer Science 430 73–87.Google Scholar
Kou, H., Liu, Y.M. and Luo, M.K. (2003). On meet-continuous dcpos. In: Zhang, G., Lawson, J.D., Liu, Y. and Luo, M. (eds.) Domain Theory, Logic and Computation, Kluwer Academic Publishers 137–149.Google Scholar
Mao, X. and Xu, L. (2006). Quasicontinuity of posets via Scott topology and sobrification. Order 23 (4) 359–369.Google Scholar
Mao, X. and Xu, L. (2009). Meet continuity properties of posets. Theoretical Computer Science 410 (42) 4234–4240.Google Scholar
Xu, L. (2006). Continuity of posets via Scott topology and sobrification. Topology and its Applications 153 (11) 1886–1894.Google Scholar
Xu, X. and Yang, J. (2009). Topological representations of distributive hypercontinuous lattices. Chinese Annals of Mathematics, Series B 30 (2) 199–206.Google Scholar
Zhang, W. and Xu, X. (2015). s 2-Quasicontinuous posets. Theoretical Computer Science 574 78–85.Google Scholar
Zhao, D. (2015). Closure spaces and completions of posets. Semigroup Forum 90 (2) 545–555.CrossRefGoogle Scholar
Zhao, D. and Fan, T. (2010). Dcpo-completion of posets. Theoretical Computer Science 411 (22) 2167–2173.CrossRefGoogle Scholar