Given a partial order $R\_{\leq}$ over a set $D$, the set of upper bounds under $R$ of a subset $S$ of $D$ is commonly defined as $\{ y \in D | \ \forall x\in S, x R y \}$.


(The set of lower bounds of $S$ may be defined as the set of upper bounds of $S$ under the converse relation $R^{-1}$)


Is there a common name for the generalization of this notion where $R$ is not a partial order, and is possibly a heterogenous relation between domain $D$ and codomain $D'$ (hence the $y$ would be elements of the codomain)? This would be a subset of the image of $S$ under $R$ (and conversely, the dual notion would be a subset of the preimage).



Name for "lower/upper bounds" of arbitrary relations?
If your relation is at all order-like, then I would recommend just staying with the upper/lower bound terminology. And unless I misunderstand you, the example you describe is actually a (strict) partial order, no? If the relation only goes from disjoint sets D to D', then this is (vacuously) transitive, irreflexive and assymetric.


But in general, every binary relation is ultimately a directed graph. In this case, you would say that y is a *common direct successor* (or *common target*) of the elements of S if x R y for all x in X. And y is a *common direct predecessor* if y R x for all x in S.


If your relation is a tree, in one direction or the other, then you could use the *common parent* and *common child* terminology.


