Complement of Bottom/Bounded Lattice
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Theorem
Let $\struct {S, \vee, \wedge, \preceq}$ be a bounded lattice.
Then the bottom $\bot$ has a unique complement, namely $\top$, top.
Proof
We know that $\bot$ is the identity for $\vee$.
Therefore, from the condition that:
- $\bot \vee a = \top$
for a complement $a$ of $\bot$, it follows that $a = \top$ is the only possibility.
Since also:
- $\bot \wedge \top = \bot$
as $\top$ is the identity for $\wedge$, the result follows.
$\blacksquare$