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

MathSciNet review:
1301499

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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 elements, whose congruence lattice Con*L* is isomorphic to *D*. We show that this result is best possible.

**[1]**George Grätzer,*General lattice theory*, Pure and Applied Mathematics, vol. 75, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**509213****[2]**George Grätzer,*On the congruence lattice of a lattice*, The Dilworth theorems, Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1990, pp. 460–464. MR**1111511****[3]**G. Grätzer and H. Lakser,*Congruence lattices of planar lattices*, Acta Math. Hungar.**60**(1992), no. 3-4, 251–268. MR**1177680**, 10.1007/BF00051643**[4]**G. Grätzer, H. Lakser, and E. T. Schmidt,*Congruence lattices of small planar lattices*, Proc. Amer. Math. Soc.**123**(1995), no. 9, 2619–2623. MR**1301498**, 10.1090/S0002-9939-1995-1301498-5**[5]**G. Grätzer and E. T. Schmidt,*On congruence lattices of lattices*, Acta Math. Acad. Sci. Hungar.**13**(1962), 179–185. MR**0139551**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1301499-7

Keywords:
Congruence lattice,
finite lattice,
distributive lattice

Article copyright:
© Copyright 1995
American Mathematical Society