# 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.

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

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 $1$.

 Title complemented lattice Canonical name ComplementedLattice Date of creation 2013-03-22 15:02:25 Last modified on 2013-03-22 15:02:25 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 26 Author CWoo (3771) Entry type Definition Classification msc 06C15 Classification msc 06B05 Synonym perspective elements Synonym complemented Related topic Perspectivity Related topic OrthocomplementedLattice Related topic PseudocomplementedLattice Related topic DifferenceOfLatticeElements Related topic Pseudocomplement Defines related elements in lattice Defines complement Defines complemented element