Given a partial order R_≤ over a set D, the set of upper bounds under R of a subset S of D is commonly defined as { y ∈ D | ∀ x ∈ 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).

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).