# Definition:Complement (Lattice Theory)

## Definition

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a bounded lattice.

Denote by $\bot$ and $\top$ the bottom and top of $S$, respectively.

Let $a \in S$.

Then $b \in S$ is called a complement of $a$ if and only if:

$b \vee a = \top$
$b \wedge a = \bot$

If $a$ has a unique complement, it is denoted by $\neg a$.

### Complemented Lattice

Suppose that every $a \in S$ admits a complement.

Then $\left({S, \vee, \wedge, \preceq}\right)$ is called a complemented lattice.

## Also denoted as

Considerably many sources use $a'$ in place of $\neg a$ to denote complement, while $\sim \! a$ is also seen.

## Linguistic Note

The word complement comes from the idea of complete-ment, it being the thing needed to complete something else.

It is a common mistake to confuse the words complement and compliment. Usually the latter is mistakenly used when the former is meant.