One observation that may save some time to the readers - at least I had to think about it for a while, being close to declare that I have found an error. In the proof of Proposition 1.81 we have $$f(p) = \bigwedge\{q \in Q| p \leq g(q)\}$$ and it is not immediately clear what happens when right hand side is empty set. From the Definition 1.60 one can deduce, that $$\bigwedge \emptyset$$ if exists must be maximal element. And indeed, if we assume that "all meets exists" we must assume also existence of the maximal element. Or am I making some mistake here?