Complemented lattice

  • 4.1 Examples of Orthokomplementen

Complementary elements

In a limited association (mathematics) V is called an element b is a complement of a if

  • And

Applies.

A limited group of which each element ( at least ) has a complement, ie complementary association.

In general there may be several elements to a complementary elements. Is the complement of a clear, then different designations are used: with subset associations ac is common for applications in which a logic ¬ in Schaltalgebren. It is

  • ¬ 0 = 1, ¬ 1 = 0

In a distributive limited association, each element can have at most one complement If a has a complement ¬ a, then ¬ a has a complement, namely:

  • ¬ (¬ A) = A.

A distributive complementary association called Boolean association or Boolean algebra.

Relative complements

If a, b elements of an association, then that means the amount determined by the a and b interval. The definition is consistent with the minor amounts to a closed interval, and it is used in the same label.

Are, then that means d relative complement of c with respect to when

  • And applies.

Again, it [ b a] can be several complementary to c elements and that of the distributive law follows the uniqueness.

An association called relatively complemented, if for every element is a relative complement in each interval.

A relatively complementary association is an association of complementary if and only if it is limited. Conversely, it must not be quite complementary, a complementary association. But it is true: A modular complementary association is relatively complemented.

Relative complements can be used for the characterization of distributive associations:

Pseudokomplemente

If a, b are two elements of an association, then it is called a greatest element c, then for the, a relative Pseudokomplement of a with respect to b.

A relative Pseudokomplement of a with respect to 0 is called simple Pseudokomplement of a

An association exists in the for each element a be an Pseudokomplement, called pseudo- complementary association.

The term for Pseudokomplemente is not uniform.

Properties

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

In a distributive lattice forms an ideal. Therefore, the existence of Pseudokomplementen in finite distributive associations is assured. The Distributivity is essential: M3 is not pseudo- complementary.

For Pseudokomplemente need not apply, even if V is distributive. However, it is always:

  • And

For Pseudokomplemente one of De Morgan's law applies:

Only applies to the dual form:

A distributive relatively - complementary association called Heyting algebra.

Orthokomplemente

In a federation, a function is called orthogonalization if it satisfies the following conditions:

  • And
  • ,

The association ( with this figure) is called ortho complementary association. orthocomplementation means of (at this orthogonalization ).

If V is a distributive complementary association, then the complement of a is also be the only possible orthocomplementation. In general, but you can also in a distributive lattice have several different Orthogonalisierungen.

Examples Orthokomplementen

  • If V is a Euclidean vector space and U1 is a subspace, then form the orthogonal vectors U to a ( possibly empty) vector space U2. U1 and U2 are Orthokomplemente in ( modular ) Association of subspaces of V.
  • The example of the Euclidean vector spaces can be generalized to arbitrary vector spaces with an inner product. Various inner products deliver this A., different Orthokomplemente the Association of subspaces of V.

References and Notes

Literature and sources

  • Helmuth Gericke: theory of associations. 2 edition. BI, Mannheim 1967.
  • Grätzer George: Lattice Theory. First concepts and distributive lattices. W.H.Freeman and Company, 1971.
  • Lattice theory
Heyting-Algebra
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