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Homomorphisms of distributive lattices as restrictions of congruences. (English) Zbl 0597.06007

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 \(\beta \phi =\psi \alpha\).

MSC:

06B10 Lattice ideals, congruence relations
06D05 Structure and representation theory of distributive lattices
06B15 Representation theory of lattices
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