Total Relation

Let $R:X\to Y\to \mathrm{Prop}$ be a relation. $R$ is considered left-total or total in $X$ iff for every $x$ in $X$, there is some $y$ s.t. $Rxy$. Likewise $R$ is right-total or total in $Y$ iff for every $y$ in $Y$, there is some $x$ s.t. $Rxy$.

This analogy extends to other relation arities.