# Complement (association theory)

## Complementary elements

In a restricted lattice (mathematics) an element is called a complement of if ${\ displaystyle b}$${\ displaystyle a}$

${\ displaystyle a \ wedge b = 0}$ and ${\ displaystyle a \ vee b = 1}$

applies.

A restricted lattice in which every element has (at least) one complement is called a complementary lattice .

In general there can be several complementary elements for one element. If the complement of is unambiguous, different terms are used: it is common for subsets , for applications in logic , for switching algebras . It applies ${\ displaystyle a}$${\ displaystyle a ^ {\ complement}}$${\ displaystyle \ neg a}$${\ displaystyle {\ bar {a}}}$

${\ displaystyle \ neg 0 = 1, \ neg 1 = 0}$.

In a distributive bounded lattice, each element can have at most one complement.
If a complement has, then also has a complement, namely ${\ displaystyle a}$${\ displaystyle \ neg a}$${\ displaystyle \ neg a}$

${\ displaystyle \ neg \ neg a = a}$.

A distributive complementary lattice is called a Boolean lattice or Boolean algebra .

## Relative complements

The non-modular association is complementary: and are both complements of . It is not relative-complementary, because the interval has no complement.${\ displaystyle N_ {5}}$${\ displaystyle b}$${\ displaystyle c}$${\ displaystyle a}$
${\ displaystyle [0, c]}$${\ displaystyle b}$

If there are elements of a lattice, then the set is called the interval determined by a and b . The definition is identical to that of a closed interval on ordered sets and the same notation is used. ${\ displaystyle a, b}$${\ displaystyle \ left \ {x \ in V \ mid a \ leq x \ land x \ leq b \ right \}}$
${\ displaystyle [a, b]}$

Are , then the relative complement of relative , if ${\ displaystyle c, d \ in [a, b]}$${\ displaystyle d}$ ${\ displaystyle c}$${\ displaystyle [a, b]}$

${\ displaystyle c \ wedge d = a}$ and applies.${\ displaystyle c \ vee d = b}$

Here, too, it applies that there can be several elements that are too complementary and that uniqueness follows from the distributive law. ${\ displaystyle [a, b]}$${\ displaystyle c}$

A lattice is called relatively complementary if there is a relative complement for every element in every interval.

A relatively complementary association is a complementary association if and only if it is restricted. Conversely, a complementary association need not be relatively complementary. However, a modular complementary lattice is relatively complementary.

Relative complements can be used to characterize distributive associations. A lattice is distributive if and only if every element has at most one relative complement in every interval.

## Pseudo complements

Rules for pseudo-complements: and can occur${\ displaystyle a <\ neg \ neg a}$${\ displaystyle \ neg a \ vee \ neg b <\ neg (a \ wedge b)}$

Are two members of a federation, then called a greatest element for which applies a relative Pseudokomplement of respect . ${\ displaystyle a, b}$${\ displaystyle c}$${\ displaystyle a \ wedge c \ leq b}$${\ displaystyle a}$${\ displaystyle b}$

A relative Pseudokomplement of respect is Pseudokomplement of . ${\ displaystyle a}$${\ displaystyle 0}$${\ displaystyle a}$

A lattice in which there is a pseudo-complement for each element is called a pseudo-complementary lattice. ${\ displaystyle a}$

The designation for pseudo complements is not uniform.

### properties

If (relative) pseudo complements exist, then they are uniquely determined.

In a distributive association forms an ideal. Hence the existence of pseudo complements in finite distributive lattices is assured. The distributivity is essential: it is not pseudocomplementary. ${\ displaystyle \ {x \ mid a \ wedge x = 0 \}}$${\ displaystyle M_ {3}}$

For pseudo complements does not have to apply, even if the lattice is distributive. But it always is ${\ displaystyle \ neg \ neg a = a}$

${\ displaystyle a \ leq \ neg \ neg a}$ and ${\ displaystyle \ neg \ neg \ neg a = \ neg a}$

One of De Morgan's laws applies to pseudo complements :

${\ displaystyle \ neg (a \ vee b) = \ neg a \ wedge \ neg b}$

For the dual form only applies:

${\ displaystyle \ neg a \ vee \ neg b \ leq \ neg (a \ wedge b)}$

A distributive relatively complementary lattice is called Heyting's algebra .

## Orthocomplements

In a lattice, a function is called orthogonalization if it meets the following conditions: ${\ displaystyle k \ colon V \ to V}$

• ${\ displaystyle a \ vee a ^ {k} = 1}$ and ${\ displaystyle a \ wedge a ^ {k} = 0}$
• ${\ displaystyle (a ^ {k}) ^ {k} = a}$
• ${\ displaystyle a \ leq b \ implies b ^ {k} \ leq a ^ {k}}$,

The lattice (with this illustration) is called the orthocomplementary lattice . is called the orthocomplement of (to this orthogonalization). ${\ displaystyle \ textstyle a ^ {k}}$${\ displaystyle a}$

If there is a distributive complementary lattice , then the complement of is also its only possible orthocomplement. In general, however, one can also have several different orthogonalizations in a distributive lattice. ${\ displaystyle V}$${\ displaystyle a}$

### Examples of orthocomplements

• If a Euclidean vector space and a sub-vector space , then the vectors which are to be orthogonal form a vector space . and are orthocomplements in the (modular) lattice of the subspaces of .${\ displaystyle V}$${\ displaystyle U_ {1}}$${\ displaystyle U}$ ${\ displaystyle U_ {2}}$${\ displaystyle U_ {1}}$${\ displaystyle U_ {2}}$${\ displaystyle V}$
• The example of the Euclidean vector spaces can be generalized to any vector spaces with an inner product . Various internal products deliver i. A. Various orthocomplements in the union of the subspaces of .${\ displaystyle V}$

These are typical examples that also led to the naming.

Examples of orthocomplements
This association allows 3 different orthogonalizations.
${\ displaystyle T_ {30}}$: The normal complement is the only possible orthocomplement.${\ displaystyle \ textstyle a ^ {\ complement} = {\ frac {30} {a}}}$
There is exactly one orthogonalization for this association.
There is no orthogonalization for .${\ displaystyle M_ {3}}$

## literature

• Gericke Helmuth: Theory of associations . 2nd Edition. BI, Mannheim 1967.
• Grätzer George: Lattice Theory. First concepts and distributive lattices . WHFreeman and Company, 1971, ISBN 978-0-486-47173-0 .

## Individual evidence

1. This follows from the reduction rule
2. G. Grätzer, Lattice theory, p. 20. In H. Gericke, Theory of Associations, p. 72 the designation is introduced differently .${\ displaystyle b / a}$
3. G. Grätzer, Lattice theory, p. 96.
4. The idea of ​​proof is that in and in each case the reduction rule does not apply, cf. H. Gericke, Theory of Associations, p. 113f${\ displaystyle N_ {5}}$${\ displaystyle M_ {5}}$
5. G. Grätzer uses a * for the pseudo complement and a * b for the relative pseudo complement (G. Grätzer, Lattice Theory: Foundation, p 99). Gericke uses a mirrored symbol for the designation. (H. Gericke, Theory of Associations, p. 119) Also or occur.${\ displaystyle \ neg}$${\ displaystyle a \ rightarrow b}$${\ displaystyle a \ Rightarrow b}$
6. H. Gericke, Theory of Associations, p. 120 f. Because of these properties, pseudo-complements can be used to model intuitionist logic .
7. H. Gericke, Theory of Associations, p. 106; however, a clearer designation is used here for the function