Let $L$ be a bounded lattice (with $0$ and $1$), and $a\in L$. A complement of $a$ is an element $b\in L$ such that

$a\land b=0$ and $a\lor b=1$.

Remark. Complements may not exist. If $L$ is a non-trivial chain, then no element (other than $0$ and $1$) has a complement. This also shows that if $a$ is a complement of a non-trivial element $b$, then $a$ and $b$ form an antichain.

An element in a bounded lattice is complemented if it has a complement. A complemented lattice is a bounded lattice in which every element is complemented.

Remarks.

• In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be related, or perspective if they have a common complement. For example, the following lattice is complemented.

  $$\xymatrix{&1\ar@{-}[ld]\ar@{-}[d]\ar@{-}[rd]&\\ a\ar@{-}[rd]&b\ar@{-}[d]&c\ar@{-}[ld]\\ &0&}$$

  Note that none of the non-trivial elements have unique complements. Any two non-trivial elements are related via the third.

• If a complemented lattice $L$ is a distributive lattice, then $L$ is uniquely complemented (in fact, a Boolean lattice). For if $y_{1}$ and $y_{2}$ are two complements of $x$, then

  $$y_{2}=1\land y_{2}=(x\lor y_{1})\land y_{2}=(x\land y_{2})\lor(y_{1}\land y_{2})=0\lor(y_{1}\land y_{2})=y_{1}\land y_{2}.$$

  Similarly, $y_{1}=y_{2}\land y_{1}$. So $y_{2}=y_{1}$.

• In the category of complemented lattices, a morphism between two objects is a $\{0,1\}$-lattice homomorphism; that is, a lattice homomorphism which preserves $0$ and $1$.