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Lim-inf convergence in partially ordered sets. (English) Zbl 1087.06004

Using standard methods of replacing the infima of residual subsets of nets by directed sets with elements eventual lower bounds, the authors study a notion of lim-inf convergence of nets in general partially ordered sets. The main result derived is that the lim-inf convergence so defined is topological if and only if the poset in question is a continuous poset. It is not difficult to see, although the authors do not point it out, that the topology generated from their notion of lim-inf convergence is the Scott topology. Closely related results may be found in Chapter II-1 of Continuous lattices and domains by G. Gierz, K. Hofmann, K. Keimel, J. Lawson, M. Mislove and D. S. Scott [Cambridge University Press, Cambridge (2003; Zbl 1088.06001)]. The authors also introduce a weaker form of lim-inf convergence and a corresponding notion of continuity of a poset and again show that convergence is topological if and only if the poset is continuous in this alternative sense.

MSC:

06B35 Continuous lattices and posets, applications

Citations:

Zbl 1088.06001
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References:

[1] Birkhoff, G., Lattice theory, Amer. math. soc. colloq. publ., 25, (1940) · Zbl 0126.03801
[2] Frink, O., Topology in lattices, Trans. amer. math. soc., 51, 569-582, (1942) · Zbl 0061.39305
[3] McShane, E.J., Order-preserving maps and integration process, Ann. of math. stud., vol. 31, (1953), Princeton Univ. Press Princeton, NJ · Zbl 0051.29301
[4] Gierz, G., A compendium of continuous lattices, (1980), Springer-Verlag Berlin · Zbl 0452.06001
[5] Lawson, J., The duality of continuous posets, Houston J. math., 5, 357-394, (1979) · Zbl 0428.06003
[6] Mathews, J.C.; Anderson, R.F., A comparison of two modes of order convergence, Proc. amer. math. soc., 18, 100-104, (1967) · Zbl 0152.20901
[7] Wolk, E.S., On order-convergence, Proc. amer. math. soc., 12, 379-384, (1961) · Zbl 0112.01904
[8] Kelley, J.L., General topology, (1955), Van Nostrand New York · Zbl 0066.16604
[9] Raney, G.N., Completely distributive lattices, Proc. amer. math. soc., 3, 677-680, (1952) · Zbl 0049.30304
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