``Complete-simple'' distributive lattices

Authors:
G. Grätzer and E. T. Schmidt

Journal:
Proc. Amer. Math. Soc. **119** (1993), 63-69

MSC:
Primary 06B15; Secondary 06B10, 06D05

MathSciNet review:
1150651

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Abstract: It is well known that the only simple distributive lattice is the two-element chain. We can generalize the concept of a simple lattice to complete lattices as follows: a complete lattice is *complete-simple* if it has only the two trivial complete congruences. In this paper we show the existence of infinite complete-simple distributive lattices.

**[1]**R. Freese, G. Grätzer, and E. T. Schmidt,*On complete congruence lattices of complete modular lattices*, Internat. J. Algebra Comput.**1**(1991), no. 2, 147–160. MR**1128008**, 10.1142/S0218196791000080**[2]**George Grätzer,*General lattice theory*, Birkhäuser Verlag, Basel-Stuttgart, 1978. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. MR**504338****[3]**G. Grätzer,*The complete congruence lattice of a complete lattice*, Lattices, semigroups, and universal algebra (Lisbon, 1988) Plenum, New York, 1990, pp. 81–87. MR**1085069****[4]**George Grätzer,*A “lattice-theoretic” proof of the independence of the automorphism group, the congruence lattice, and the subalgebra lattice of an infinitary algebra*, Algebra Universalis**27**(1990), no. 4, 466–473. MR**1387894**, 10.1007/BF01188992**[5]**G. Grätzer, P. Johnson, and E. T. Schmidt,*A representation of*-*algebraic lattices*, manuscript.**[6]**G. Grätzer and H. Lakser,*On complete congruence lattices of complete lattices*, Trans. Amer. Math. Soc.**327**(1991), no. 1, 385–405. MR**1036003**, 10.1090/S0002-9947-1991-1036003-5**[7]**G. Grätzer and H. Lakser,*On congruence lattices of 𝔪-complete lattices*, J. Austral. Math. Soc. Ser. A**52**(1992), no. 1, 57–87. MR**1137597****[8]**G. Grätzer, H. Lakser, and B. Wolk,*On the lattice of complete congruences of a complete lattice: on a result of K. Reuter and R. Wille*, Acta Sci. Math. (Szeged)**55**(1991), no. 1-2, 3–8. MR**1124939****[9]**E. Tamás Schmidt,*Congruence relations of algebraic structures*, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl.**9**(1959), 163–174 (Hungarian). MR**0114783****[10]**Klaus Reuter and Rudolf Wille,*Complete congruence relations of concept lattices*, Acta Sci. Math. (Szeged)**51**(1987), no. 3-4, 319–327. MR**940936****[11]**S. K. Teo,*Every finite lattice is a complete congruence lattice*, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.**33**(1990), 177–182 (1991). MR**1139362**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1150651-0

Keywords:
Complete lattice,
distributive lattice,
complete congruence,
congruence lattice

Article copyright:
© Copyright 1993
American Mathematical Society