×

Infinite distributive laws versus local connectedness and compactness properties. (English) Zbl 1190.54022

The author deals with a large array of distributivity properties for lattices and connects these to an equally large array of local connectedness and compactness properties for frames. The latter properties are meaningful only in the class of \(T_0\)-spaces, where the distributivity is applied to specialization order – \(x\leq y\) means \(x\in\text{cl}\{y\}\) – or to the lattice of open sets. There are no explicit examples in the paper, which makes it hard to get a feeling for the differences between the properties and to see what they are actually good for.
Reviewer: K. P. Hart (Delft)

MSC:

54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54D05 Connected and locally connected spaces (general aspects)
54D45 Local compactness, \(\sigma\)-compactness
06D10 Complete distributivity
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alexandroff, P., Diskrete Räume, Mat. sb. (N.S.), 2, 501-518, (1937) · Zbl 0018.09105
[2] Banaschewski, B., Essential extensions of \(T_0\)-spaces, Appl. gen. topol., 7, 233-246, (1977) · Zbl 0371.54026
[3] Bandelt, H.-J.; Erné, M., Representations and embeddings of \(\mathcal{M}\)-distributive lattices, Houston J. math., 10, 315-324, (1984) · Zbl 0551.06014
[4] Birkhoff, G., Lattice theory, Amer. math. soc. colloq. publ., vol. 25, (1973), Amer. Math. Soc. Providence, RI · Zbl 0126.03801
[5] Crawley, P.; Dilworth, R.P., Algebraic theory of lattices, (1973), Prentice Hall, Inc. · Zbl 0494.06001
[6] Erné, M., Scott convergence and Scott topologies on partially ordered sets II, (), 61-96
[7] Erné, M., Distributivgesetze und die dedekindsche schnittvervollständigung, Abh. braunschw. wiss. ges., 33, 117-145, (1982)
[8] Erné, M., Convergence and distributivity: A survey, () · Zbl 1067.06003
[9] Erné, M., On the existence of decompositions in lattices, Algebra universalis, 16, 338-343, (1983) · Zbl 0516.06004
[10] Erné, M., The ABC of order and topology, (), 57-83 · Zbl 0735.18005
[11] Erné, M., The dedekind – macneille completion as a reflector, Order, 8, 159-173, (1991) · Zbl 0738.06004
[12] Erné, M., Bigeneration in complete lattices and principal separation in ordered sets, Order, 8, 197-221, (1991) · Zbl 0738.06005
[13] Erné, M., Algebraic ordered sets and their generalizations, (), 113-192 · Zbl 0791.06007
[14] Erné, M., \(\mathcal{Z}\)-continuous posets and their topological manifestation, Appl. categ. structures, 7, 31-70, (1999) · Zbl 0939.06005
[15] Erné, M., The polarity between approximation and distribution, (), 173-210 · Zbl 1061.06010
[16] Erné, M., Minimal bases, ideal extensions, and basic dualities, Topology proc., 29, 445-489, (2005) · Zbl 1128.06001
[17] Erné, M., Choiceless, pointless, but not useless: dualities for categories of preframes, Appl. categ. structures, 15, 541-572, (2007) · Zbl 1137.06001
[18] M. Erné, Strong local properties of hyperweak spaces, preprint, 2008
[19] Erné, M.; Gatzke, H., Convergence and continuity in partially ordered sets and semilattices, (), 9-40
[20] Erné, M.; Kopperman, R., Natural continuity space structures on dual Heyting algebras, Fund. math., 136, 157-177, (1990) · Zbl 0727.06012
[21] Erné, M.; Pultr, A.; Sichler, J., Closure frames and web spaces, (2000), Charles University Prague, KAM—DIMATIA Series 2000-501, DAM and ITI
[22] Erné, M.; Wilke, G., Standard completions for quasiordered sets, Semigroup forum, 27, 351-376, (1983) · Zbl 0517.06009
[23] Fraïssé, R., Theory of relations, Stud. logic found. math., vol. 118, (1986), North-Holland Amsterdam · Zbl 0593.04001
[24] Gierz, G.; Hofmann, K.H.; Keimel, K.; Lawson, J.D.; Mislove, M.; Scott, D.S., A compendium of continuous lattices, (1980), Springer Berlin · Zbl 0452.06001
[25] Gierz, G.; Hofmann, K.H.; Keimel, K.; Lawson, J.D.; Mislove, M.; Scott, D.S., Continuous lattices and domains, Encyclopedia math. appl., vol. 93, (2003), Cambridge University Press
[26] Gierz, G.; Lawson, J., Generalized continuous lattices and hypercontinuous lattices, Rocky mountain J. math., 11, 271-296, (1981) · Zbl 0472.06014
[27] Gierz, G.; Lawson, J.D.; Stralka, A.R., Quasicontinuous posets, Houston J. math., 9, 191-208, (1983) · Zbl 0529.06002
[28] Grätzer, G., General lattice theory, (1978), Birkhäuser-Verlag Basel · Zbl 0385.06015
[29] Higman, G., Ordering by divisibility in abstract algebras, Proc. London math. soc., 3, 326-336, (1952) · Zbl 0047.03402
[30] Hoffmann, R.-E., Topological spaces admitting a “dual”, (), 157-166
[31] Hofmann, K.H.; Lawson, J., On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom, J. austral. math. soc. ser. A, 36, 194-212, (1984) · Zbl 0536.06005
[32] Hofmann, K.H.; Mislove, M., Local compactness and continuous lattices, (), 209-248
[33] Howard, P.; Rubin, J.E., Consequences of the axiom of choice, Math. surveys monogr., vol. 59, (1998), Amer. Math. Soc. Providence, RI · Zbl 0947.03001
[34] Isbell, J.R., Function spaces and adjoints, Math. scand., 36, 317-339, (1975) · Zbl 0309.54016
[35] Johnstone, P.T., Stone spaces, (1982), Cambridge University Press Cambridge · Zbl 0499.54001
[36] Kříž, I.; Pultr, A., A spatiality criterion and a quasitopology which is not a topology, Houston J. math., 15, 215-233, (1989) · Zbl 0695.54002
[37] Lawson, J.D., \(T_0\)-spaces and pointwise convergence, Topology appl., 21, 73-76, (1985) · Zbl 0575.54013
[38] Levine, N., Strongly connected sets in topology, Amer. math. monthly, 72, 1098-1101, (1965) · Zbl 0134.42105
[39] Mao, X.; Xu, L., Quasicontinuity of posets via Scott topology and sobrification, Order, 23, 359-369, (2006) · Zbl 1124.06002
[40] Papert, S., Which distributive lattices are lattices of closed sets?, Math. proc. Cambridge philos. soc., 55, 172-176, (1959) · Zbl 0178.33703
[41] Raney, G.N., Completely distributive complete lattices, Proc. amer. math. soc., 3, 677-680, (1952) · Zbl 0049.30304
[42] Raney, G.N., A subdirect-union representation for completely distributive complete lattices, Proc. amer. math. soc., 4, 518-522, (1953) · Zbl 0053.35201
[43] Schwarz, F.; Weck, S., Scott topology, isbell topology, and continuous convergence, (), 251-272 · Zbl 0598.54005
[44] Steen, L.A.; Seebach, J.A., Counterexamples in topology, (1968), Holt, Rinehart and Winston, Inc. New York · Zbl 0211.54401
[45] Tarski, A., Sur LES classes closes par rapport á certaines opérations élémentaires, Fund. math., 16, 181-305, (1929) · JFM 56.0843.03
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.