Definition:Maximal Chain

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $\left({T, \preceq}\right) \subseteq \left({S, \preceq}\right)$ be a chain in $\left({S, \preceq}\right)$ such that there is no other chain in $\left({S, \preceq}\right)$ which has $\left({T, \preceq}\right)$ as a proper subset.

Then $\left({T, \preceq}\right)$ is a maximal chain in $S$.