Definition:Order Completion

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

An ordered set $\left({T, \preceq_T}\right)$ is an order completion of $S$ if and only if:

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

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