Noun
semilattice (plural semilattices)
English Wikipedia has an article on:semilatticeWikipedia
(mathematics) A partially ordered set that either has a join (a least upper bound) for any nonempty finite subset (a join-semilattice or upper semilattice) or has a meet (or greatest lower bound) for any nonempty finite subset (a meet-semilattice or lower semilattice). Equivalently, an underlying set which has a binary operation which is associative, commutative, and idempotent.
As mentioned, becomes graded by this semilattice. Source: Internet
Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as morphisms. Source: Internet
On the other hand, some authors have no use for this distinction of morphisms (especially since the emerging concepts of "complete semilattice morphisms" can as well be specified in general terms). Source: Internet
Given a homomorphism from an arbitrary semigroup to a semilattice, each inverse image is a (possibly empty) semigroup. Source: Internet