When studying inverse-semigroups, idempotent elements play an important role, and semilattices occur naturally as sets of idempotent elements that commute with each other. I have the impression that semilattices are closely related to lattices, but the following remark on wikipedia makes me wonder how well this relation is really understood (or taught):
Regrettably, it is often the case that standard treatments of lattice theory define a semilattice, if that, and then say no more.
I also wonder how the following statement from a similar question should be interpreted:
any meet semilattice of finite height is also a join semilattice
The following two Hasse diagrams show that this statement is only valid if the semilattice is bounded (i.e. if the underlying semigroup has an identity element):
In the left diagram, the meet of $a$ and $b$ is $0$, but their join doesn't exist. Now adjoining an identity element to a semigroup is easy (and canonical), and we can see in the right diagram that this fixes the problem.
However, what really interests me is how to canonically "invent" all missing joins in the infinite case, similar to the way we can adjoin an identity element to fix the finite case. Would it help if we were given a (compatible) topology or a (compatible) topological uniform structure?
Can the Dedekind–MacNeille completion (which is the smallest complete lattice that contains the given partial order) help to clarify the relation between lattice and semilattice theory?