TY - JOUR AU - Sun, Tao AU - Li, Qingguo AU - Fan, Nianbai PY - 2019 TI - \(O_1\)-convergence in partially ordered sets JO - Journal of Nonlinear Sciences and Applications SP - 634--643 VL - 12 IS - 10 AB - 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. SN - ISSN 2008-1901 UR - http://dx.doi.org/10.22436/jnsa.012.10.02 DO - 10.22436/jnsa.012.10.02 ID - Sun2019 ER - TY - BOOK TI - Lattice Theory AU - G. Birkhoff PB - American Mathematical Society PY - 1940 DA - 1940// CY - New York ID - Birkhoff1940 ER - TY - BOOK TI - An introduction to axiomatic set theory (in chinese) AU - M. M. Dai AU - H. Y. Chen AU - D. W. Zheng PB - Science Press PY - 2011 DA - 2011// CY - Beijing ID - Dai2011 ER - TY - BOOK TI - Introduction to Lattices and Order AU - B. A. Davey AU - H. A. Priestley PB - Cambridge University Press PY - 2002 DA - 2002// CY - Cambridge ID - Davey2002 ER - TY - BOOK TI - General Topology AU - R. Engelking PB - PWN-Polish Scientific Publishers PY - 1977 DA - 1977// CY - Warszawa ID - Engelking1977 ER - TY - JOUR TI - Topology in lattice AU - O. Frink JO - Trans. Amer. Math. Soc. PY - 1942 DA - 1942// VL - 51 ID - Frink1942 ER - TY - BOOK TI - Continuous Lattices and Domain AU - G. Grierz AU - K. H. Hofmann AU - K. Keimel AU - J. D. Lawson AU - M. W. Mislove AU - D. S. Scott PB - Camberidge University Press PY - 2003 DA - 2003// CY - Camberidge ID - Grierz2003 ER - TY - BOOK TI - General Topology AU - J. L. Kelly PB - Van Nostrand PY - 1955 DA - 1955// CY - New York ID - Kelly1955 ER - TY - JOUR TI - A comparison of two modes of order convergence AU - J. C. Mathews AU - R. F. Anderson JO - Proc. Amer. Math. Soc. PY - 1967 DA - 1967// VL - 18 ID - Mathews1967 ER - TY - BOOK TI - Order--Preserving Maps and Integration Processes AU - E. J. Mcshane PB - Princeton University Press PY - 1953 DA - 1953// CY - Princeton ID - Mcshane1953 ER - TY - JOUR TI - Order Convergence and Order Topology on a Poset AU - V. Olejček JO - Internat. J. Theoret. Phys. PY - 1999 DA - 1999// VL - 38 ID - Olejček1999 ER - TY - JOUR TI - Strongly compactly atomistic orthomodular lattices and modular ortholattices AU - Z. Riečanová JO - Tatra Mt. Math. Publ. PY - 1998 DA - 1998// VL - 15 ID - Riečanová1998 ER - TY - JOUR TI - Birkhoff's order--convergence in partially ordered sets AU - T. Sun AU - Q. G. Li AU - L. K. Guo JO - Topology Appl. PY - 2016 DA - 2016// VL - 207 ID - Sun2016 ER - TY - JOUR TI - Some further result on order--convergence in posets AU - K. Y. Wang AU - B. Zhao JO - Topology Appl. PY - 2013 DA - 2013// VL - 160 ID - Wang2013 ER - TY - JOUR TI - On order--convergence AU - E. S. Wolk JO - Proc. Amer. Math. Soc. PY - 1961 DA - 1961// VL - 12 ID - Wolk1961 ER - TY - JOUR TI - A note on continuous partially ordered sets AU - H. Zhang JO - Semigroup Forum PY - 1993 DA - 1993// VL - 47 ID - Zhang1993 ER - TY - JOUR TI - The double Scott topology on a lattice AU - D. S. Zhao JO - Chin. Ann. Math. Ser. A PY - 1989 DA - 1989// VL - 10 ID - Zhao1989 ER - TY - JOUR TI - Order topology and bi-Scott topology on poset AU - B. Zhao AU - K. Y. Wang JO - Acta Math. Sin. (Engl. Ser.) PY - 2011 DA - 2011// VL - 27 ID - Zhao2011 ER - TY - JOUR TI - Lim--inf--convergence in partially ordered sets AU - B. Zhao AU - D. S. Zhao JO - J. Math. Anal. Appl. PY - 2005 DA - 2005// VL - 309 ID - Zhao2005 ER - TY - JOUR TI - Order--convergence and Lim--inf$_{\mathcal{M}}$--convergence in poset AU - Y. H. Zhou AU - B. Zhao JO - J. Math. Anal. Appl. PY - 2007 DA - 2007// VL - 325 ID - Zhou2007 ER -