Suppose I have a lattice, and I pick one of its elements $a$. Then I define a relation on all elements of the lattice by $$x \sim_a y \iff x \vee a = y \vee a.$$ In other words, two elements are related exactly when they have the same join with $a$.
(For example, in the divisibility lattice on the positive integers, $6\sim_{10} 15$ because $\mathrm{lcm}(6,10)=\mathrm{lcm}(15,10)$. )
Is there a name for this equivalence relation? It seems like a natural enough notion to me, but I am having trouble finding a reference to it anywhere. I suppose this could be because (a) there is something problematic about it that I have not realized, and as a result it is not something people consider or (b) there is an equivalent way of thinking about it that is more natural and so I'm just thinking about it in the wrong terms.
Any pointers would be greatly appreciated.