\begin{definition}[Definition:Convex Set (Order Theory)/Definition 1]
A subset $A$ of an ordered set $\left({S, \preceq}\right)$ is '''convex (in $S$)''' {{iff}}:
:$\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$
\end{definition}