Grätzer, George; Lakser, Harry Homomorphisms of distributive lattices as restrictions of congruences. (English) Zbl 0597.06007 Can. J. Math. 38, 1122-1134 (1986). Let L be a finite lattice, and let I be an ideal of L. The map \(\psi\) : \(\theta\) \(\to \theta_ I\), restricting a congruence relation \(\theta\) to I, is a 0 and 1 preserving homomorphism of Con L (the congruence lattice of L) into Con I. We prove the converse: Theorem. Let D and E be finite distributive lattices, and let \(\phi\) : \(D\to E\) be a 0 and 1 preserving homomorphism of D into E. Then there exists a sectionally complemented finite lattice L, and an ideal I of L, such that there are isomorphisms \(\alpha\) : \(D\to Con L\), \(\beta\) : \(E\to Con I\), satisfying \(\beta \phi =\psi \alpha\). Cited in 2 ReviewsCited in 8 Documents MSC: 06B10 Lattice ideals, congruence relations 06D05 Structure and representation theory of distributive lattices 06B15 Representation theory of lattices Keywords:finite distributive lattice; congruence lattice; sectionally complemented finite lattice; ideal PDFBibTeX XMLCite \textit{G. Grätzer} and \textit{H. Lakser}, Can. J. Math. 38, 1122--1134 (1986; Zbl 0597.06007) Full Text: DOI