# Definition:Order Completion

## Definition

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

An ordered set $\struct {T, \preceq_T}$ 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
$(3):\quad \struct {T, \preceq_T}$ is a complete ordered set
$(4):\quad$ For all ordered sets $\struct {T', \preceq_{T'} }$ satisfying $(1), (2)$ and $(3)$, there is a unique increasing injection $\phi: T' \to T$