User:Dfeuer/Order Completion is Minimal
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Note: our current definition of completion is as a superset. We presumably want also a definition as an order embedding. This theorem will of course take on a slightly different form in that case.
Theorem
Let $\struct {T, \le}$ be a complete lattice.
Let $S \subseteq T$.
Suppose that $T$ is an order completion of $S$.
Then for any $U$ such that $S \subseteq U \subsetneqq T$,
- $\struct {U, \le}$ is not complete.
Proof
Aiming for a contradiction, suppose $U$ is complete.