Definition:Convex Set (Order Theory)/Definition 1

Definition
A subset $A$ of an ordered set $\left({S, \preceq}\right)$ is convex (in $S$) :
 * $\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$

Also see

 * Equivalence of Definitions of Convex Set (Order Theory)