Geodesic
Homomorphisms of Distributive Lattices as Restrictions of Congruences
Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1122-1134

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Given a lattice L and a convex sublattice K of L, it is well-known that the map Con L → Con K from the congruence lattice of L to that of K determined by restriction is a lattice homomorphism preserving 0 and 1. It is a classical result (first discovered by R. P. Dilworth, unpublished, then by G. Grätzer and E. T. Schmidt [2], see also [1], Theorem II.3.17, p. 81) that any finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. Although it has been conjectured that any algebraic distributive lattice is the congruence lattice of some lattice, this has not yet been proved in its full generality. The best result is in [4]. The conjecture is true for ideal lattices of lattices with 0; see also [3]. 
Grätzer, George; Lakser, Harry. Homomorphisms of Distributive Lattices as Restrictions of Congruences. Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1122-1134. doi: 10.4153/CJM-1986-056-2
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[1] 1. Grätzer, G., General lattice theory, Pure and Applied Mathematics Series (Academic Press, New York, N.Y.). Google Scholar

[2] 2. Grätzer, G. and Schmidt, E. T., On congruence lattices of lattices, Acta Math, Acad. Sci. Hungar. 13 (1962), 179–185. Google Scholar

[3] 3. Pudlák, P., On congruence lattices of lattices. To appear in Algebra Universalis. Google Scholar

[4] 4. Schmidt, E. T., The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice, preprint. Google Scholar

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