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Connectedness Properties of Lattices

Published online by Cambridge University Press:  20 November 2018

R. Vainio
R. Vainio*
Affiliation:
Department of MathematicsÅbo Akademi SF-20500 Åbo 50, Finland
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Abstract

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Let L be a lattice and q a convergence structure (or a topology) finer than the interval topology of L. In case of compact maximal chains and continuous lattice translations, the connected components of the space (L,q) are characterized using lattice conditions only. Moreover, lattice conditions of L are related to connectedness conditions of the order convergence space (L, o). Throughout this note, maximal chain conditions and maximal chain techniques play an important role.

Keywords

54A2054D0554H12Maximal chains in latticesconvergence structures (topologies) on latticesorder convergenceinterval topologyconnected components
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 3 , 01 September 1986 , pp. 314 - 320
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Ball, R. N., Convergence and Cauchy structures on lattice ordered groups, Trans. Amer. Math. Soc. 259(1980), pp. 357–392.Google Scholar
2. Erné, M., On the relativization of chain topologies, Pacific J. Math. 84 (1979), pp. 43 — 52.Google Scholar
3. Erné, M., Separation axioms for interval topologies, Proc. Amer. Math. Soc. 79 (1980), pp. 185—190.Google Scholar
4. Erné, M., Scott convergence and Scott topology in partially ordered sets II. in Lecture Notes in Math. 871. Springer-Verlag, Berlin-Heidelberg-New York 1981.Google Scholar
5. Erné, M., and Week, S., Order convergence in lattices, Rocky Mountain J. Math. 10 (1980), pp. 805–818.Google Scholar
6. Gähler, W., Grundstrukturen der Analysis I—II, Akademie-Verlag und Birkhäuser Verlag, 1977—78.Google Scholar
7. Kent, D. C., On the order topology in a lattice, Illinois J. Math. 10 (1966), pp. 90–96.Google Scholar
8. Rennie, B. C., Lattices, Proc. London Math. Soc. 52 (1951), pp. 386–400.Google Scholar
9. Vainio, R., The limit space CC(X, Y) and the connected components of X and Y, Math. Nachr. 113 (1983), pp. 29–37.Google Scholar
10. Vainio, R., On connectedness in limit space theory, Proceedings of the Schwerin 1983 School on Convergence Structures, to appear.Google Scholar
11. Ward, L. E., Partially ordered topological spaces, Proc. Amer. Math. Soc. 5 (1954), pp. 144—161.Google Scholar