Given a lattice provided with a join-irreducible representation of its elements, there is a natural "intersection" operator $A \mathbin{\dot\cap} B$ that returns the join of the setwise intersection of the join-irreducible representations of $A$ and $B$.

Is there a standard name for this operator?

Note that this is very different from the meet. Indeed, representing all elements using the lower sets of their join-irreducible representation, the meet is the intersection of the lower sets, whereas $\dot\cap$ is given by the lower set generated by the intersetion of the join-irreducible representations.