Is there an established name for a map of complete lattices $f : L \to L'$ that preserves nonempty suprema? I.e. for all $U \subseteq L$ with $U \neq \emptyset$, $$ f( \bigvee U) = \bigvee_{u \in U} f(u)$$
An example of such a map is $a \mapsto a \lor c$.
I have heard this property being called 'continuous' but I would like to find a reference. Another name for preserving the empty suprema is 'strictness', since this means that $f(\bot) = \bot$ (note that $a \mapsto a \lor c$ is not strict).
I'm mainly interested in the case where we have a binary operation $\ast : L \times L \to L$ that preserves non-empty suprema in each argument.
(A structure $(Q, \ast : Q \times Q \to Q)$ where $Q$ is a complete lattice and $\ast$ preserves all suprema in each argument is known as a quantale.)
Thank you!