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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On complete congruence lattices of complete lattices
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by G. Grätzer and H. Lakser PDF
Trans. Amer. Math. Soc. 327 (1991), 385-405 Request permission

Abstract:

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In this paper, we characterize this lattice as a complete lattice. In other words, for a complete lattice $L$, we construct a complete lattice $K$ such that $L$ is isomorphic to the lattice of complete congruence relations of $K$. Regarding $K$ as an infinitary algebra, this result strengthens the characterization theorem of congruence lattices of infinitary algebras of G. Grätzer and W. A. Lampe. In addition, we show how to construct $K$ so that it will also have a prescribed automorphism group.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 327 (1991), 385-405
  • MSC: Primary 06B10; Secondary 06A23, 08A30
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1036003-5
  • MathSciNet review: 1036003