Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be *incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$

Consider the following statement:

If $L$ contains an ideal $J$ such that $\bigvee_L J \notin J$, then there is a complete lattice $K$ and an incomplete lattice homomorphism $f: L\to K$.

Is this true? (The converse is true.)