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:



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Is this true? (The converse is true.)



Incomplete lattice homomorphisms between complete lattices (2)
Yes. Suppose that $J$ is an ideal in $L$ such that $\bigvee^{L}J\not\in J$. Let $\mathcal{Id}(L)$ be the set of all ideals of $L$. Then $\mathcal{Id}(L)$ is a complete lattice. Define a mapping $V:L\rightarrow\mathcal{Id}(L)$ by letting $V(x)=\downarrow x$ for each $x\in L$. Then $V$ is a lattice homomorphism. However, since $\bigvee^{X}J\not\in J$, we have $\bigvee V[J]=J$, but $V[\bigvee J]\neq J$, so $V$ is an incomplete lattice homomorphism.


