I am wondering if I am understanding this concept correctly.
A partially ordered set $S$ is called a meet-semilattice if, for all $x,y \in S$, a unique greatest lower bound of $x$ and $y$ exists.
So let's say we have the set $S = \{1,2,3,12,36\}$, and we want to ask if this set, when partially ordered by divisibility, is a meet-semilattice.
I am inclined to say that NO it is not. If we take $x = 12$ and $y=36$, then there are three lower bounds ($1,2,3$). But, none of them is the greatest since $2$ and $3$ do not divide each other.
Is my logic/approach correct?
The set $\{12,36\}$ actually has $4$ lower bounds; the one you missed is $12$, which also happens to be the greatest lower bound. In general, if $x$ and $y$ are comparable, they automatically have a greatest lower bound (and a least upper bound). You only need to check incomparable pairs, in this case, $\{2,3\}$.