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?
Best Answer
I will now try to give an answer based on my current understanding.
For the finite case, the Dedekind–MacNeille completion of a meet semilattice obviously reduces to the adjunction of a maximal element (if it was missing). This is also true for a meet semilattice of finite height, because a lattice of finite height is necessarily complete.
For the general infinite case, the Dedekind–MacNeille completion will invent more elements than required, because not every lattice is a complete lattice. Even worse, "the completion of a distributive lattice need not itself be distributive, and the completion of a modular lattice may not remain modular". But even if we could prove the existence of a unique minimal lattice containing a given semilattice, this still wouldn't give a satisfactory relation between lattice and semilattice theory. Therefore, a more promising route is to manually translate the notions from lattice theory to semilattice theory, like it is done for distributivity and modularity.
This doesn't seem to help. One could try to investigate whether the Dedekind–MacNeille completion allows to define a canonical topology on the completion based on the given topology (and similarly for topological uniform structures).
This impression could be a hasty generalization from the cases where there is a known close relationship (completeness, distributivity and modularity). Since complete lattices/semilattices are so common, this relationship might be all that is needed, but this doesn't answer the question whether semilattices and lattices are essentially the same objects (similar to monoids and semigroups), or whether there are fundamental differences between the two. I have come to believe that there really are such differences, for example that the minimal semilattice containing a given partially ordered set is easy to construct, while it is unclear how to construct a corresponding minimal lattice.