Complement of Bottom/Bounded Lattice

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.

$\blacksquare$

Also see

• Complement of Top