可数S_2-拟连续偏序集
|
摘要
作为广义可数逼近偏序集与S2-拟连续偏序集的共同推广,引入了可数S2-拟连续偏序集的概念并讨论了它的一些性质.本文的主要结果:(1)可数S2-拟连续偏序集上的可数way below关系满足插入性质;(2)可数S2-拟连续偏序集关于其上的弱σ-Scott拓扑为局部紧致的可数sober空间;(3)偏序集P为可数S2-连续偏序集当且仅当P为可数S2-交连续的可数S2-拟连续偏序集.
As a common generalization of generalized countably approximating posets and S2-quasicontinuous posets, the concept of countably S2-quasicontinuous posets is introduced and some properties of them are discussed. The main results of this paper are following:(1) The countabe way below relation on a countably S2-quasicontinuous posets has the interpolation property;(2) A countably S2-quasicontinuous poset is a locally compact countably sober space with respect to itsσ-Scott topology;(3) A poset is countably S2-continuous iff it is countably meet S2-continuous and countably S2-quasicontinuous.
As a common generalization of generalized countably approximating posets and S2-quasicontinuous posets, the concept of countably S2-quasicontinuous posets is introduced and some properties of them are discussed. The main results of this paper are following:(1) The countabe way below relation on a countably S2-quasicontinuous posets has the interpolation property;(2) A countably S2-quasicontinuous poset is a locally compact countably sober space with respect to itsσ-Scott topology;(3) A poset is countably S2-continuous iff it is countably meet S2-continuous and countably S2-quasicontinuous.
引文
[1]Gierz G,Hofmann K,Keimel K,et al.Continuous Lattices and Domains[M].Cambridge:Cambridge University Press,2003.
[2]Ern′e M.Scott convergence and Scott topology on partially order sets[A].In:B.Banaschewski and R.-E.Hoffman,eds.,Continuous Lattices,Bremen 1979,Lecture Notes in Math.871[C].Springer-Verlag,Berlin-Heidelberg-New York,1981,61-96.
[3]Zhao Dongsheng.Semicontinuous Lattices[J].Algebra Universalis,1997,37(4):458-476.
[4]Ho Weng Kin,Zhao Dongsheng.Lattices of Scott-closed sets[J].Comment Math Univ Carolin,2009,50(2):297-314.
[5]Lee S O.On countably approximating lattices[J].J Korean Math Soc,1988,25(1):11-23.
[6]Han Y H,Hong S S,Lee C K,et al.A generalization of continuous posets[J].Comm Korean Math Soc,1989,4(1):129-138.
[7]Yang Jinbo,Liu Min.On generalized countably approximating posets[J].J Chungcheong Math Soc,2012,23(4):415-424.
[8]Zhang Wenfeng,Xu Xiaoquan.S2-quasicontinuous posets[J].Theoret Comput Sci,2015,572:78-85.
[9]Ern′e M.The Dedekind-Mac Neille completion as a reflector[J].Order,1991,8(2):159-173.
[10]Xu Xiaoquan,Yang Jinbo.Topological representations of distributive hypercontinuous lattices[J].Chin Ann Math,2009,30B(2):199-206.
[11]Yang Jinbo,Shi Junmiao.Countably sober spaces[J].Elect Notes Theoret Comput Sci,2017,333:143-151.
[2]Ern′e M.Scott convergence and Scott topology on partially order sets[A].In:B.Banaschewski and R.-E.Hoffman,eds.,Continuous Lattices,Bremen 1979,Lecture Notes in Math.871[C].Springer-Verlag,Berlin-Heidelberg-New York,1981,61-96.
[3]Zhao Dongsheng.Semicontinuous Lattices[J].Algebra Universalis,1997,37(4):458-476.
[4]Ho Weng Kin,Zhao Dongsheng.Lattices of Scott-closed sets[J].Comment Math Univ Carolin,2009,50(2):297-314.
[5]Lee S O.On countably approximating lattices[J].J Korean Math Soc,1988,25(1):11-23.
[6]Han Y H,Hong S S,Lee C K,et al.A generalization of continuous posets[J].Comm Korean Math Soc,1989,4(1):129-138.
[7]Yang Jinbo,Liu Min.On generalized countably approximating posets[J].J Chungcheong Math Soc,2012,23(4):415-424.
[8]Zhang Wenfeng,Xu Xiaoquan.S2-quasicontinuous posets[J].Theoret Comput Sci,2015,572:78-85.
[9]Ern′e M.The Dedekind-Mac Neille completion as a reflector[J].Order,1991,8(2):159-173.
[10]Xu Xiaoquan,Yang Jinbo.Topological representations of distributive hypercontinuous lattices[J].Chin Ann Math,2009,30B(2):199-206.
[11]Yang Jinbo,Shi Junmiao.Countably sober spaces[J].Elect Notes Theoret Comput Sci,2017,333:143-151.