Definition:Complement (Lattice Theory)

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

Denote with $\bot$ and $\top$ the identities for $\vee$ and $\wedge$, respectively.

Let $a \in S$.

Then $b \in S$ is called a complement of $a$ iff:


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

By Complement is Unique, there is at most one such $b$.

Thus one speaks of the complement of $a$, and denotes it with $\neg a$.

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

Also see

 * Complemented Lattice, a bounded lattice in which every element has a complement