In a pre-order prec (or a category) one can speak of initial objects 0, or terminal objects 1, meaning that 0precx for all x --- (or 0rightarrow!x ) --- which also gives the notion of a universal object under several. E.g., among objects preceding both of b1,b2, with the restricted relation (a1,a2)|a1preca2,aiprecbj 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.
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