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.