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

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 ConK into ConL. Extending a result of A. 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. 1991 Mathematical Subject Classification. Primary 06B10; Secondary 06D05.