A note on ?small representations of finite distributive lattices as congruence lattices?

Order ◽  
1996 ◽  
Vol 13 (4) ◽  
pp. 365-367 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Distributive Lattices ◽  
Congruence Lattices ◽  
Finite Distributive Lattices

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

2000 ◽  
Vol 68 (1) ◽  
pp. 85-103 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser ◽  
E. T. Schmidt
Keyword(s):  
Distributive Lattices ◽  
Irreducible Elements ◽  
Finite Lattices ◽  
Congruence Lattices ◽  
Finite Distributive Lattices

AbstractLet 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.

scholarly journals The Central Decomposition of FD01(n)

Order ◽  
2021 ◽  
Author(s):  
Peter Köhler
Keyword(s):  
Distributive Lattices ◽  
Finitely Generated ◽  
Finite Distributive Lattices

AbstractThe paper presents a method of composing finite distributive lattices from smaller pieces and applies this to construct the finitely generated free distributive lattices from appropriate Boolean parts.

scholarly journals Congruences on a Distributive Lattice

1954 ◽  
Vol 10 (2) ◽  
pp. 76-77
Author(s):  
H. A. Thueston
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Distributive Lattices ◽  
The Subject ◽  
The Many ◽  
Finite Distributive Lattices

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.

About polytopes of valuations on finite distributive lattices

Order ◽  
1985 ◽  
Vol 2 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Hans Dobbertin
Keyword(s):  
Distributive Lattices ◽  
Finite Distributive Lattices

scholarly journals On the relation of a distributive lattice to its lattice of ideals

1972 ◽  
Vol 7 (3) ◽  
pp. 377-385 ◽  
Author(s):  
Herbert S. Gaskill
Keyword(s):  
Distributive Lattice ◽  
Related Result ◽  
Distributive Lattices ◽  
Relationship Of ◽  
The Relationship ◽  
Finite Distributive Lattices ◽  
Lattice Of Ideals

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

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