Congruence-preserving extensions

of finite lattices

to sectionally complemented lattices

Authors:
G. Grätzer and E. T. Schmidt

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1903-1915

MSC (1991):
Primary 06B10; Secondary 08A05

DOI:
https://doi.org/10.1090/S0002-9939-99-04671-7

Published electronically:
March 3, 1999

MathSciNet review:
1476133

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that *every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice*.

**1.**G. Grätzer,*General Lattice Theory*, Pure and Applied Mathematics**75**, Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New York-London; Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. Birkhäuser Verlag, Basel-Stuttgart; Akademie Verlag, Berlin, 1978. xiii+381 pp. (Expanded Second Edition, 1998.) MR**80c:06001b****2.**G. Grätzer, H. Lakser, and E. T. Schmidt,*Isotone maps as maps of congruences. I. Abstract maps*, Acta Math. Acad. Sci. Hungar.**75**(1997), 81-111. CMP**97:10****3.**-,*Representing isotone maps as maps of congruences. II. Concrete maps*, manuscript.**4.**-,*Congruence representations of join homomorphisms of distributive lattices: A short proof*, Math. Slovaca**46**(1996), 363-369. CMP**98:02****5.**-,*Restriction of standard congruences on lattices*, manuscript. Accepted for publication in Contributions to General Algebra 10, Proceedings of the Klagenfurt Conference May 29-June 1, 1997. Edited by D. Dorninger, E. Eigenthaler, H. J. Kaiser, H. Kautschitsch, W. More and W. B. Müller. B. G. Teubner, Stuttgart. Aug. 1997.**6.**G. Grätzer and E. T. Schmidt,*On congruence lattices of lattices*, Acta Math. Acad. Sci. Hungar.**13**(1962), 179-185. MR**25:2983****7.**-,*A lattice construction and congruence-preserving extensions*, Acta Math. Hungar.**66**(1995), 275-288. MR**95m:06018****8.**-,*The Strong Independence Theorem for automorphism groups and congruence lattices of finite lattices*, Beiträge Algebra Geom.**36**(1995), 97-108. MR**96h:06014****9.**O. Ore,*Theory of equivalence relations*, Duke Math. J.**9**(1942), 573-627. MR**4:128f****10.**M. Plo\v{s}\v{c}ica, J. T\r{u}ma, and F. Wehrung,*Congruence lattices of free lattices in non-distributive varieties*, Colloq. Math.**76**(1998), 269-278.**11.**P. Pudlák and J. T\r{u}ma,*Every finite lattice can be embedded into a finite partition lattice*, Algebra Universalis**10**(1980), 74-95. MR**81e:06013****12.**M. Tischendorf,*The representation problem for algebraic distributive lattices*, Ph. D. thesis, Fachbereich Mathematik der Technischen Hochschule Darmstadt, Darmstadt, 1992. MR**95g:06010**

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Additional Information

**G. Grätzer**

Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Email:
gratzer@cc.umanitoba.ca

**E. T. Schmidt**

Affiliation:
Mathematical Institute of the Technical University of Budapest, Műegyetem rkp. 3, H-1521 Budapest, Hungary

Email:
schmidt@math.bme.hu

DOI:
https://doi.org/10.1090/S0002-9939-99-04671-7

Keywords:
Congruence lattice,
congruence-preserving embedding,
sectionally complemented lattice,
finite

Received by editor(s):
July 16, 1996

Received by editor(s) in revised form:
September 22, 1997

Published electronically:
March 3, 1999

Additional Notes:
The research of the first author was supported by the NSERC of Canada.

The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. T7442.

Communicated by:
Lance W. Small

Article copyright:
© Copyright 1999
American Mathematical Society