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

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

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

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