Order, lattices, ordered algebraic structures Definition

Complemented lattice

Definition

Let $L$ be a bounded lattice with least element $0$ , greatest element $1$ , and join and meet $\vee$ and $\wedge$ . A complement of an element $a \in L$ is an element $b \in L$ such that $a \wedge b = 0 \quad\text{and}\quad a \vee b = 1.$ The lattice $L$ is complemented if every element of $L$ has at least one complement.

Remarks

Example: the lattice of subspaces of a finite-dimensional vector space $V$ . For a subspace $U$ , extend a basis of $U$ to a basis of $V$ and let $W$ be the span of the added vectors; then $U \cap W = \set{0}$ and $U + W = V$ .

Non-example: the chain $0 < a < 1$ . The element $a$ has no complement: $a \wedge x = 0$ forces $x = 0$ , and then $a \vee x = a \ne 1$ .

In the diamond lattice M3 each of the three middle elements has the other two as complements, so complements are not unique. In a distributive lattice an element has at most one complement, and a complemented distributive lattice is a Boolean algebra.

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Tag	`T08FF77H16`
URI	`m2://T08FF77H16`
Kind	definition
Writing	W0 stub
MSC	06C15, 06B05
Defines	Complemented lattice
Uses	Bounded lattice, Join and meet

Complemented lattice

Definition

Remarks

Reading path

Further reading

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