In a reading group, we are going through the B. Fong and D. Spivak book "An Invitation to Applied Category Theory, Seven Sketches in Compositionality", and we are in a proposition that states: Let $\mathcal P=(P,\leq)$ be a preorder. It has all joins iff it has all meets. And, for instance, the following preorder has all joins but not all the meets.
The next is a screenshot of the proof, and we think the set $M_A$ could be the empty set, so what may be wrong?
Thanks for any help.
You are correct that for your poset $P$ that the subset $A = \{a, b \}$ doesn't have a meet, and that the set $M_A$ is the empty set.
This is precisely where the problem occurs: your poset doesn't have all joins because it doesn't have a join of the empty set. If you check through the definition for what a join of the empty set ought to mean, working through the vacuous conditions, you'll see that a join of the empty set is a bottom element. Which the poset $P$ doesn't have.
Therefore there's no contradiction in there being no meet of $\{a, b \}$, because the poset doesn't have all joins.
