of a partially ordered set
A subset containing with any two elements $a$ and $b$ the entire interval $[a,b]$ (cf. Interval and segment).
A definition not involving the notion of interval is: A subset $A$ of a partially ordered set is convex if $a\leq b\leq c$ and $a,c\in A$ imply $b\in A$.
In the real line (with its usual ordering) the convex subsets are exactly the connected subsets (for the usual topology). This need not hold for more general ordered topological spaces. However, if a partially ordered set is equipped with the interval topology (cf. Order topology), then its connected subsets are convex.
Convex subset. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Convex_subset&oldid=31462