order-convex set

An order-convex set is a subset of a poset such that any element in the poset that is between two elements in the subset is also in the subset.

Formally,
an order-convex set $I$ is a subset of poset $P$ such that for all $x,y\in I$ and $z\in P$ with $x\le z\le y$, then $z\in I$.

Every interval is an order-convex set.