Locally modular lattices and locally distributive lattices
HTML articles powered by AMS MathViewer
- by Shûichirô Maeda PDF
- Proc. Amer. Math. Soc. 44 (1974), 237-243 Request permission
Abstract:
A locally modular (resp. locally distributive) lattice is a lattice with a congruence relation and each of whose equivalence class has sufficiently many elements and is a modular (resp. distributive) sublattice. Both the lattice of all closed subspaces of a locally convex space and the lattice of projections of a locally finite von Neumann algebra are locally modular. The lattice of all ${T_1}$-topologies of an infinite set is locally distributive.References
- R. W. Bagley, On the characterization of the lattice of topologies, J. London Math. Soc. 30 (1955), 247–249. MR 67862, DOI 10.1112/jlms/s1-30.2.247
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- Otto Fröhlich, Das Halbordnungssystem der topologischen Räume auf einer Menge, Math. Ann. 156 (1964), 79–95 (German). MR 166750, DOI 10.1007/BF01359926
- G. Grätzer and E. T. Schmidt, Standard ideals in lattices, Acta Math. Acad. Sci. Hungar. 12 (1961), 17–86 (English, with Russian summary). MR 133265, DOI 10.1007/BF02066675
- Roland E. Larson and W. J. Thron, Covering relations in the lattice of $T_{1}$-topologies, Trans. Amer. Math. Soc. 168 (1972), 101–111. MR 296883, DOI 10.1090/S0002-9947-1972-0296883-6
- F. Maeda and S. Maeda, Theory of symmetric lattices, Die Grundlehren der mathematischen Wissenschaften, Band 173, Springer-Verlag, New York-Berlin, 1970. MR 0282889
- A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc. 122 (1966), 379–398. MR 190893, DOI 10.1090/S0002-9947-1966-0190893-2
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 44 (1974), 237-243
- MSC: Primary 06A30
- DOI: https://doi.org/10.1090/S0002-9939-1974-0340132-4
- MathSciNet review: 0340132