We mention joins above as if the objects are join-semilattices; one can just as well consider them meet-semilattices and say that the homomorphisms preserve finitary meets (including the top element). For in itself, this is purely a difference in notational convention. However, this choice corresponds to using either of two inclusion functors representing as a subcategory of Pos.
is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so has all the usual properties of such categories. The free semilattice on a set is the finite powerset of , that is the set of finite subsets of , thought of as a join-semilattice. (Note that this exists even in predicative mathematics, as long as we are allowed to define sets by recursion over natural numbers, although you have to construct it by general nonsense instead of as a subset of the full powerset. For purposes of constructive mathematics, by ‘finite’ we mean Kuratowski finite.)