# Definition:Complement (Lattice Theory)

## Definition

Let $\struct {S, \vee, \wedge, \preceq}$ 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 $\struct {S, \vee, \wedge, \preceq}$ 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.

## Also see

- Definition:Complemented Lattice, a bounded lattice in which every element has a complement
- Complement in Distributive Lattice is Unique

## 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.