×
Sun, Tao; Li, Qingguo; Fan, Nianbai

\(\mathcal{MN}\)-convergence and \(\lim\)-\(\inf_{\mathcal{M}}\)-convergence in partially ordered sets. (English) Zbl 1410.54010

Open Math. 16, 1077-1090 (2018).
Summary: In this paper, we first introduce the notion of \(\mathcal{MN}\)-convergence in posets as a unified form of \(O\)-convergence and \(O_{2}\)-convergence. Then, by studying the fundamental properties of \(\mathcal{MN}\)-topology which is determined by \(\mathcal{MN}\)-convergence according to the standard topological approach, an equivalent characterization to the \(\mathcal{MN}\)-convergence being topological is established. Finally, the \(\lim\)-\(\inf_{\mathcal{M}}\)-convergence in posets is further investigated, and a sufficient and necessary condition for \(\lim\)-\(\inf_{\mathcal{M}}\)-convergence to be topological is obtained.

Cited in 1 Document

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
06A06 Partial orders, general

Keywords:

\(\mathcal{MN}\)-convergence; \(\mathcal{MN}\)-topology; \(\lim\)-\(\inf_{\mathcal{M}}\)-convergence; \(\mathcal{M}\)-topology
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Birkhoff G., Lattice Theory,1940, American Mathematical Society Colloquium Publications.; Birkhoff, G., Lattice Theory (1940) · Zbl 0063.00402
[2] Frink O., Topology in lattice, Trans. Am. Math. Soc., 1942, 51, 569-582.; Frink, O., Topology in lattice, Trans. Am. Math. Soc., 51, 569-582 (1942) · Zbl 0061.39305
[3] Mcshane E.J., Order-Preserving Maps and Integration Process, 1953, Princeton, Princeton University Press.; Mcshane, E. J., Order-Preserving Maps and Integration Process (1953) · Zbl 0051.29301
[4] Zhao B., Zhao D.S., Lim-inf-convergence in partially ordered sets, J. Math. Anal. Appl., 2005, 309, 701-708.; Zhao, B.; Zhao, D. S., Lim-inf-convergence in partially ordered sets, J. Math. Anal. Appl., 309, 701-708 (2005) · Zbl 1087.06004
[5] Wang K.Y., Zhao B., Some further results on order-convergence in posets, Topol. Appl., 2013, 160, 82-86.; Wang, K. Y.; Zhao, B., Some further results on order-convergence in posets, Topol. Appl., 160, 82-86 (2013) · Zbl 1280.06003
[6] Zhao B., Li J., \(O_2\)-convergence in posets, Topol. Appl., 2006, 153, 2971-2975.; Zhao, B.; Li, J., \(O_2\)-convergence in posets, Topol. Appl., 153, 2971-2975 (2006) · Zbl 1096.06006
[7] Li Q.G., Zou Z.Z., A result for \(O_2\)-convergence to be topological in posets, Open Math., 2016, 14, 205-211.; Li, Q. G.; Zou, Z. Z., A result for \(O_2\)-convergence to be topological in posets, Open Math., 14, 205-211 (2016) · Zbl 1348.54005
[8] Zhou Y.H., Zhao B., Order-convergence and lim-\(inf_𝓜\)-convergence in posets, J. Math. Anal. Appl., 2007, 325, 655-664.; Zhou, Y. H.; Zhao, B., Order-convergence and lim-\(inf_𝓜\)-convergence in posets, J. Math. Anal. Appl., 325, 655-664 (2007) · Zbl 1116.06007
[9] Venugopalan P., Z-continuous posets, Houston J. Math., 1996, 12, 275-294.; Venugopalan, P., Z-continuous posets, Houston J. Math., 12, 275-294 (1996) · Zbl 0614.06007
[10] Kelly J.L., General Topology, 1955, New York, Van Nostrand Group.; Kelly, J. L., General Topology (1955) · Zbl 0066.16604
[11] Mathews J.C., Anderson R.F., A comparison of two modes of order convergence, Proc. Am. Math. Soc., 1967, 18(1), 100-104.; Mathews, J. C.; Anderson, R. F., A comparison of two modes of order convergence, Proc. Am. Math. Soc., 18, 1, 100-104 (1967) · Zbl 0152.20901
[12] Wolk E.S., On order-convergence, Proc. Am. Math. Soc., 1961, 12(3), 379-384.; Wolk, E. S., On order-convergence, Proc. Am. Math. Soc., 12, 3, 379-384 (1961) · Zbl 0112.01904
[13] Zhao D.S., The double Scott topology on a lattice, Chin. Ann. Math., Ser. A., 1989, 10(2), 187-193.; Zhao, D. S., The double Scott topology on a lattice, Chin. Ann. Math., Ser. A., 10, 2, 187-193 (1989) · Zbl 0679.06003
[14] Zhao B., Wang K.Y., Order topology and bi-Scott topology on a poset, Acta Math. Sin. Engl. Ser., 2011, 27, 2101-2106.; Zhao, B.; Wang, K. Y., Order topology and bi-Scott topology on a poset, Acta Math. Sin. Engl. Ser., 27, 2101-2106 (2011) · Zbl 1242.06009
[15] Engelking R., General Topology, 1977, Warszawa, Polish Scientific Publishers.; Engelking, R., General Topology (1977) · Zbl 0373.54002
[16] Zhang H., A note on continuous partially ordered sets, Semigroup Forum, 1993, 47, 101-104.; Zhang, H., A note on continuous partially ordered sets, Semigroup Forum, 47, 101-104 (1993) · Zbl 0777.06007
[17] Davey B.A., Priestley H.A., Introduction to Lattices and Order, 2002, Cambridge, Cambridge University Press.; Davey, B. A.; Priestley, H. A., Introduction to Lattices and Order (2002) · Zbl 1002.06001
[18] Grierz G., Hofman K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S., Continuous Lattices and Domain, 2003, Cambridge, Camberidge University Press.; Grierz, G.; Hofman, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domain (2003) · Zbl 1088.06001
[19] Grierz G., Hofman K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S., A Compendium of Continuous Lattices, 1980, Berlin: Springer-Verlag Press.; Grierz, G.; Hofman, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., A Compendium of Continuous Lattices (1980) · Zbl 0452.06001
[20] Olejček V., Order convergence and order topology on a poset, Int. J. Theor. Phys., 1999, 38, 557-561.; Olejček, V., Order convergence and order topology on a poset, Int. J. Theor. Phys., 38, 557-561 (1999) · Zbl 0931.54027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.