Those two operations are called "meet" and "join", and in principle they work with any linear subspaces (point, line, plane, etc.) So the meet of a line and a plane in 3D is a point, the join of a point and a line in 3D is a plane, etc.

Wikipedia is good on this stuff if you are patient with mathematician's way of expressing things...

https://en.m.wikipedia.org/wiki/Join_and_meet

https://en.m.wikipedia.org/wiki/Linear_subspace

"Lattice of subspaces

The operations intersection and sum make the set of all subspaces a bounded modular lattice, where the {0} subspace, the least element, is an identity element of the sum operation, and the identical subspace V, the greatest element, is an identity element of the intersection operation."