order theory - Name for "lower/upper bounds" of arbitrary relations?

In a pre-order $prec$ (or a category) one can speak of initial objects $0$, or terminal objects $1$, meaning that $0prec x$ for all $x$ --- (or $0rightarrow^! x$ ) --- which also gives the notion of a universal object under several. E.g., among objects preceding both of $b_1,b_2$, with the restricted relation ${(a_1,a_2)|a_1prec a_2 ,a_iprec b_j}$ one can talk again about maximal objects and terminal objects, either of which notions might make a sensible candidate for "greatest lower bound" in this setting.

If you're not assuming the relation is transitive, you might want to take a (possibly graded category) transitive closure, or look at "transitive neighborhoods", or even just immediate neighborhoods as suggesed by Joel David Hamkins.

Of course, this is all quite speculative; I've not done any work where this notion was wanted.