Given a relation $R \subseteq S \times T$,
what is the concept that $\forall s \in S$, there is at least a $t \in T$, s.t. $(s,t) \in R$? ("left-surjective"?)
what is the concept that $\forall t \in T$, there is at least a $s \in S$, s.t. $(s,t) \in R$? ("right-surjective"?)
Thanks.
These are called "totality" and "surjectivity," respecitively.
The latter generalizes the terminology for functions. The former generalizes the terminology for partial functions: "function" is the same as "partial function which happens to be total."
Finally, although you didn't ask about it, the remaining pair of notions are called "functionality" and "injectivity," respectively: $R$ is functional iff for each $x$ there is at most one $y$ such that $xRy$ (equivalently, functional relations are exactly partial functions), and $R$ is injective iff for each $y$ there is at most one $x$ such that $xRy$.