\(O_1\)-convergence in partially ordered sets
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Authors
Tao Sun
- College of Mathematics and Physics, Hunan University of Arts and Science, Changde, Hunan 415000, P. R. China.
- College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China.
Qingguo Li
- College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China.
Nianbai Fan
- College of Computer Science and Electronic Engineering, Hunan University, Changsha, Hunan 410082, P. R. China.
Abstract
Based on the introduction of notions of \(S^*\)-doubly continuous posets and
B-topology in [T. Sun, Q. G. Li, L. K. Guo, Topology Appl., \(\bf207\) (2016), 156--166], in this paper, we further propose the
concept of B-consistent \(S^*\)-doubly
continuous posets and prove that the
\(O_1\)-convergence in a poset is topological if and only if the poset is a
B-consistent \(S^*\)-doubly continuous poset. This is the main result which
can be seen as a sufficient and necessary condition for the \(O_1\)-convergence in
a poset being topological. Additionally, in order to present natural examples
of posets which satisfy such condition, several special sub-classes of
B-consistent \(S^*\)-doubly continuous posets are investigated.
Share and Cite
ISRP Style
Tao Sun, Qingguo Li, Nianbai Fan, \(O_1\)-convergence in partially ordered sets, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 10, 634--643
AMA Style
Sun Tao, Li Qingguo, Fan Nianbai, \(O_1\)-convergence in partially ordered sets. J. Nonlinear Sci. Appl. (2019); 12(10):634--643
Chicago/Turabian Style
Sun, Tao, Li, Qingguo, Fan, Nianbai. "\(O_1\)-convergence in partially ordered sets." Journal of Nonlinear Sciences and Applications, 12, no. 10 (2019): 634--643
Keywords
- \(O_1\)-convergence
- B-topology
- \(S^*\)-doubly continuous poset
- B-consistent \(S^*\)-doubly continuous poset
MSC
References
-
[1]
G. Birkhoff, Lattice Theory, American Mathematical Society, New York (1940)
-
[2]
M. M. Dai, H. Y. Chen, D. W. Zheng, An introduction to axiomatic set theory (in chinese), Science Press, Beijing (2011)
-
[3]
B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge (2002)
-
[4]
R. Engelking, General Topology, PWN-Polish Scientific Publishers, Warszawa (1977)
-
[5]
O. Frink, Topology in lattice, Trans. Amer. Math. Soc., 51 (1942), 569--582
-
[6]
G. Grierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domain, Camberidge University Press, Camberidge (2003)
-
[7]
J. L. Kelly, General Topology, Van Nostrand, New York (1955)
-
[8]
J. C. Mathews, R. F. Anderson, A comparison of two modes of order convergence, Proc. Amer. Math. Soc., 18 (1967), 100--104
-
[9]
E. J. Mcshane, Order--Preserving Maps and Integration Processes, Princeton University Press, Princeton (1953)
-
[10]
V. Olejček, Order Convergence and Order Topology on a Poset, Internat. J. Theoret. Phys., 38 (1999), 557--561
-
[11]
Z. Riečanová, Strongly compactly atomistic orthomodular lattices and modular ortholattices, Tatra Mt. Math. Publ., 15 (1998), 143--153
-
[12]
T. Sun, Q. G. Li, L. K. Guo, Birkhoff's order--convergence in partially ordered sets, Topology Appl., 207 (2016), 156--166
-
[13]
K. Y. Wang, B. Zhao, Some further result on order--convergence in posets, Topology Appl., 160 (2013), 82--86
-
[14]
E. S. Wolk, On order--convergence, Proc. Amer. Math. Soc., 12 (1961), 379--384
-
[15]
H. Zhang, A note on continuous partially ordered sets, Semigroup Forum, 47 (1993), 101--104
-
[16]
D. S. Zhao, The double Scott topology on a lattice, Chin. Ann. Math. Ser. A, 10 (1989), 187--193
-
[17]
B. Zhao, K. Y. Wang, Order topology and bi-Scott topology on poset, Acta Math. Sin. (Engl. Ser.), 27 (2011), 2101--2106
-
[18]
B. Zhao, D. S. Zhao, Lim--inf--convergence in partially ordered sets, J. Math. Anal. Appl., 309 (2005), 701--708
-
[19]
Y. H. Zhou, B. Zhao, Order--convergence and Lim--inf$_{\mathcal{M}}$--convergence in poset, J. Math. Anal. Appl., 325 (2007), 655--664