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Small representations of finite distributive lattices as congruence lattices


Authors: George Grätzer, Ivan Rival and Nejib Zaguia
Journal: Proc. Amer. Math. Soc. 123 (1995), 1959-1961
MSC: Primary 06B10; Secondary 06D05
DOI: https://doi.org/10.1090/S0002-9939-1995-1301499-7
Correction: Proc. Amer. Math. Soc. 126 (1998), 2509-2510.
MathSciNet review: 1301499
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Abstract | References | Similar Articles | Additional Information

Abstract: A recent result of G. Grätzer, H. Lakser, and E. T. Schmidt states that for any distributive lattice D with n join-irreducible elements, there exists a lattice L with $ O({n^2})$ elements, whose congruence lattice ConL is isomorphic to D. We show that this result is best possible.


References [Enhancements On Off] (What's this?)

  • [1] G. Grätzer, General lattice theory, Pure Appl. Math. Ser., Academic Press, New York, 1978; Mathematische Reihe, Band 52, Birkhäuser Verlag, Basel; Akademie Verlag, Berlin. MR 509213 (80c:06001b)
  • [2] -, Results on the congruence lattice of a lattice, The Dilworth Theorems. Selected Papers of Robert P. Dilworth (K. P. Bogart, R. Freese, and J. P. S. Kung, eds.), Birkhäuser Verlag, Basel-Boston, 1990, pp. 460-464. MR 1111511
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  • [4] G. Grätzer, H. Lakser, and E. T. Schmidt, Congruence lattices of small planar lattices. Proc. Amer. Math. Soc. (to appear). MR 1301498 (95k:06017a)
  • [5] G. Grätzer and E. T. Schmidt, On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13 (1962), 179-185. MR 0139551 (25:2983)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1301499-7
Keywords: Congruence lattice, finite lattice, distributive lattice
Article copyright: © Copyright 1995 American Mathematical Society

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