complemented lattice

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.
  
  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 $\lbrace 0,1\rbrace$ -lattice homomorphism; that is, a lattice homomorphism which preserves $0$ and $1$ .