Are ideals of an sup semilattice always non-empty?

Question

I am trying to do an exercise from the book A Compendium of Continuous Lattices.

Exercise: Let $L$ be a set with a transitive relation, and let $A,B$ be ideals of $L$.

(i) $A\cap B$ is an ideal of $L$ iff $A\cap B \ne \emptyset$, for $L$ a sup-semilattice.

In the book an ideal is defined as a directed lower set without stating that the set should be non-empty, so the emptyset should also be an ideal because it fullfills the definition vacuously. Are there any restrictions if $L$ is a sup semilattice (i.e. a poset in which any finite nonempty subset has a supremum) such that the $\emptyset$ could not be an ideal? Otherweise the exercise makes no sense to me.

Answer 1

Look at the first complete paragraph on page $2$, at the definition of the set $L$ being directed:

As the empty subset is included in this definition as a finite subset, $L$ must be nonempty.

In order for $L$ to be directed, every finite subset of $L$ must have an upper bound in $L$, and $\varnothing$ is a finite subset of $L$, so $\varnothing$ has some upper bound in $L$, which is therefore non-empty.

Since an ideal is a directed set, it must by definition be non-empty. (That’s also why they have to define $\operatorname{Id}_0L=\operatorname{Id}L\cup\{\varnothing\}$.)