\begin{definition}[Definition:Quasiuniformity]
Let $S$ be a set.
A '''quasiuniformity''' on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the following '''quasiuniformity axioms''':
{{:Definition:Quasiuniformity Axioms}}
That is, a '''quasiuniformity''' on $S$ is a filter on the cartesian product $S \times S$ (from $(\text U 1)$ to $(\text U 3)$) which also fulfils the condition:
:$\forall u \in \UU: \exists v \in \UU$ such that whenever $\tuple {x, y} \in v$ and $\tuple {y, z} \in v$, then $\tuple {x, z} \in u$
which can be seen to be an equivalent statement to $(\text U 4)$.
$u \circ v$ in this context can be seen to be equivalent to composition of relations.
Thus a '''quasiuniformity''' on $S$ is a filter on $S \times S$ which also fulfils the condition that every element is the composition of another element with itself.
\end{definition}