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Congruence Representations of Join-homomorphisms of Finite Distributive Lattices: Size and Breadth

Part of: Boolean algebras (Boolean rings) Distributive lattices Lattices

Published online by Cambridge University Press:  09 April 2009

G. Grätzer ,
H. Lakser  and
E. T. Schmidt
G. Grätzer
Affiliation:
Department of Mathematics University of ManitobaWinnipeg, Man. R3T 2N2, Canada
H. Lakser
Affiliation:
Department of Mathematics University of ManitobaWinnipeg, Man. R3T 2N2, Canada
E. T. Schmidt
Affiliation:
Mathematical Institute Technical University of BudapestMũegyetem rkp. 3 H-1521 Budapest, Hungary
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Abstract

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Let K and L be lattices, and let ϕ be a homomorphism of K into L.Then ϕ induces a natural 0-preserving join-homomorphism of Con K into Con L.

Extending a result of Huhn, the authors proved that if D and E are finite distributive lattices and ψ is a 0-preserving join-homomorphism from D into E, then D and E can be represented as the congruence lattices of the finite lattices K and L, respectively, such that ψ is the natural 0-preserving join-homomorphism induced by a suitable homomorphism ϕ: K → L. Let m and n denote the number of join-irreducible elements of D and E, respectively, and let k = max (m, n). The lattice L constructed was of size O(22(n+m)) and of breadth n+m.

We prove that K and L can be constructed as ‘small’ lattices of size O(k5) and of breadth three.

Keywords

Latticefinitecongruencedistributivebreadth

MSC classification

Secondary: 06B10: Ideals, congruence relations 06D05: Structure and representation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 1 , February 2000 , pp. 85 - 103
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Duffus, D., Jónsson, B. and Rival, I., ‘Structure results for function lattices’, Canad. J. Math. 33 (1978), 392–400.Google Scholar
[2]Funayama, N. and Nakayama, T., ‘On the distributivity of a lattice of lattice-congruences’, Proc. Imp. Acad. Tokyo 18 (1942), 553–554.Google Scholar
[3]Grätzer, G., General lattice theory, 2nd ed. (Birkhäauser Basel, 1998).Google Scholar
[4]Grätzer, G. and Lakser, H., ‘Congruence lattices of planar lattices’, Acta Math. Hungar. 60 (1992), 251–268.CrossRefGoogle Scholar
[5]Grätzer, G., Lakser, H. and Schmidt, E. T., ‘Congruence lattices of small planar lattices’, Proc. Amer. Math. Soc. 123 (1995), 2619–2623.Google Scholar
[6]Grätzer, G., Lakser, H. and Schmidt, E. T., ‘Congruence representations of join homomorphisms of distributive lattices: A short proof’, Math. Slovaca 46 (1996), 363–369.Google Scholar
[7]Grätzer, G., Lakser, H. and Schmidt, E. T., ‘Isotone maps as maps of congruences. I. Abstract maps’, Acta Math. Acad. Sci. Hungar 75 (1997), 81–111.Google Scholar
[8]Grätzer, G., Rival, I. and Zaguia, N., ‘Small representations of finite distributive lattices as congruence lattices’, Proc. Amer Math. Soc. 123 (1995), 1959–1961.Google Scholar
[9]Grätzer, G. and Schmidt, E. T., ‘On congruence lattices of lattices’, Acta Math. Acad. Sci. Hungar 13 (1962), 179–185.CrossRefGoogle Scholar
[10]Grätzer, G. and Wang, D., ‘A lower bound for congruence representations’, Order 14 (1997), 67–74.CrossRefGoogle Scholar
[11]Huhn, A. P., ‘On the representation of distributive algebraic lattices. I-III’, Acta Sci. Math. (Szeged) 45 (1983), 239–246; 53 (1989), 3–10, 11–18.Google Scholar
[12]Schmidt, E. T., ‘Zur Charakterisierung der Kongruenzverbande der Verbände’, Mat. asopis Človen. Akad. Vied. 18 (1968), 3–20.Google Scholar
[13]Teo, S.-K., ‘On the length of the congruence lattice of a lattice’, Period. Math. Hungar 21 (1990), 179–186.CrossRefGoogle Scholar
[14]Zhang, Y., ‘A note on ‘Small representations of finite distributive lattices as congruence lattices‘’, Order 13 (1996), 365–367.Google Scholar