Complement of Top/Bounded Lattice

Theorem

Let $\struct {S, \vee, \wedge, \preceq}$ be a bounded lattice.

Then the top $\top$ has a unique complement, namely $\bot$, bottom.

Proof

By Dual Pairs (Order Theory), $\top$ is dual to $\bot$.

The result follows from the Duality Principle and Complement of Bottom.

$\blacksquare$

Also see

• Complement of Bottom