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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The lattice of closed congruences on a topological lattice


Author: Dennis J. Clinkenbeard
Journal: Trans. Amer. Math. Soc. 263 (1981), 457-467
MSC: Primary 06B30
MathSciNet review: 594419
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0594419-9
Keywords: Brouwerian lattice, congruence lattice, compact Hausdorff topological lattice
Article copyright: © Copyright 1981 American Mathematical Society