  We prove the following result: Let K be a lattice, let D be a distributive
lattice with zero, and let $\phi$: Con K $\to$ D be a {&#8744;,
0}-homomorphism, where Conc K denotes the {&#8744;, 0}-semilattice of all
&#64257;nitely generated congruences of K. Then there are a lattice L, a
lattice homomorphism f : K $\to$ L, and an isomorphism $\ga$: Conc L $\to$ D
such that $\ga$ Conc f = $\phi$. Furthermore, L and f satisfy many additional
properties, for example: (i) L is relatively complemented. (ii) L has
de&#64257;nable principal congruences. (iii) If the range of $\phi$ is
co&#64257;nal in D, then the convex sublattice of L generated by f[K] equals L.
We mention the following corollaries, that extend many results obtained in the
last decades in that area: -- Every lattice K such that Conc K is a lattice
admits a congruence-preserving extension into a relatively complemented
lattice. -- Every {&#8744;, 0}-direct limit of a countable sequence of
distributive lattices with zero is isomorphic to the semilattice of compact
congruences of a relatively complemented lattice with zero.
