Definition:Order Completion

Definition
Let $\left({S, \preceq_S}\right)$ be a poset.

A poset $\left({T, \preceq_T}\right)$ is said to be an order completion of $S$ iff:


 * $(1):\quad S \subseteq T$
 * $(2):\quad \preceq_T \restriction_S \, = \, \preceq_S$, where $\restriction$ denotes restriction
 * $(3):\quad \left({T, \preceq_T}\right)$ is a complete poset
 * $(4):\quad$ For all posets $\left({T', \preceq_{T'}}\right)$ satisfying $(1), (2)$ and $(3)$, there is a unique order-preserving injection $\phi: T' \to T$

Also see

 * Order Completion Unique up to Isomorphism