The lattice of closed congruences on a topological lattice
HTML articles powered by AMS MathViewer
- by Dennis J. Clinkenbeard
- Trans. Amer. Math. Soc. 263 (1981), 457-467
- DOI: https://doi.org/10.1090/S0002-9947-1981-0594419-9
- PDF | Request permission
Abstract:
Our primary objectives are: (1) if $L$ is a lattice endowed with a topology making both the meet and join continuous then (i) the natural map which associates a congruence with the smallest topologically closed congruence containing it preserves finite meets and arbitrary joins; (ii) the lattice of such closed congruences is a complete Brouwerian lattice; (2) if $L$ is a topological (semi) lattice with the unit interval as a (semi) lattice homomorphic image then the lattice of closed (semi) lattice congruences has no compatible Hausdorff topology.References
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053 P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewood Cliffs, N. J., 1973.
- E. E. Floyd, Boolean algebras with pathological order topologies, Pacific J. Math. 5 (1955), 687–689. MR 73563
- Karl H. Hofmann and Michael W. Mislove, The lattice of kernel operators and topological algebra, Math. Z. 154 (1977), no. 2, 175–188. MR 444534, DOI 10.1007/BF01241832
- Karl Heinrich Hofmann, Michael Mislove, and Albert Stralka, The Pontryagin duality of compact $\textrm {O}$-dimensional semilattices and its applications, Lecture Notes in Mathematics, Vol. 396, Springer-Verlag, Berlin-New York, 1974. MR 0354921
- Albert R. Stralka, The congruence extension property for compact topological lattices, Pacific J. Math. 38 (1971), 795–802. MR 304259
Bibliographic Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 263 (1981), 457-467
- MSC: Primary 06B30
- DOI: https://doi.org/10.1090/S0002-9947-1981-0594419-9
- MathSciNet review: 594419