Definition:Maximal Chain

Let $$\left({S; \preceq}\right)$$ be a poset.

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 $$S$$.