A semilattice is like a directed lattice in either
the meet (LCM) or join (GCD) direction ("for any nonempty finite subset").

Meet-semilattices are "lower semilattices" whereas join-semilattices are "upper
semilattices."

A set S partially ordered by the binary relation ≤ is a meet-semilattice if
For all elements x and y of S, the greatest lower bound of the set {x, y} exists.

References

https://en.wikipedia.org/wiki/Semilattice