Definition:Maximal Chain

Definition
Let $\struct {S, \preceq}$ be an ordered set.

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

Then $\struct {T, \preceq}$ is a maximal chain in $S$.