Distributive lattice
A distributive lattice is a special structure of mathematics. Compared to general associations in which operations and only the associative laws that Kommutativgesetze and the absorption laws are required for the two (two -digit ), distributive laws apply in a distributive lattice additionally for both directions.
The validity of the distributive makes organizations more interesting. They can be examined easily, as occurring terms are easier to reshape and there are "good" representations. This distributive lattices are very common, even in areas outside of mathematics. Boolean algebras are special distributive lattices.
Clarification
In the following, we mean by the " Association V " always the dressing.
An association is called distributive lattice if and only if for all:
- (D1)
- (D2)
One can derive each of the two statements from the others with the help of the Association of axioms. Therefore, it is sufficient to require the validity of these two distributive laws.
Each distributive association is modular, but not vice versa.
A modular dressing which is not distributive, always contains the association, the association of the subgroups of the Klein four-group, as a sublattice. This results in the "Test":
Examples
Distributive lattices can be found in many areas inside and outside of mathematics. Distributive lattices are:
- Any totally ordered set
- For every natural number n is the amount T_n its divisors with the divisibility as order relation (ie gcd and lcm as links )
- Lots of association and
- Every Boolean algebra
- The open sets of a topological space with the order
Reduction rule
In a distributive lattice the reduction rule: Apply to the two equations
- And follows.
The example shows that this rule does not apply in any associations. It is available in the following sense typical of distributive lattices:
Complements in distributive associations
For a given element a of a limited association is called an element b with the property
- And
A complement of a
While generally a elements may be several complementary elements, the following applies:
One calls a uniquely determined complement of with or (especially in applications in logic ) or.
A distributive lattice in which every element has a ( uniquely determined ) complement, ie, Boolean algebra.
Even in a non- distributive lattice, each element can have exactly one complement. Thus one can infer the Distributivity, you have to ask for more:
If V is a distributive lattice and have complements, then also and complements and it is
- And
This is a different formulation of de Morgan 's laws.
Representation theorem for distributive lattices
Distributive lattices are also to characterize different because Birkhoff (1933 ) and Stone ( 1936) have shown:
Of course it follows that every distributive association can be embedded in a Boolean algebra.
Other properties
Each sublattice of a distributive lattice is distributive, whereas partial associations are not always distributive.
The homomorphic image of a distributive lattice is distributive.
The direct product of any number of distributive associations is distributive.
Complete distributivity
An association called - volldistributiv applies if for any choice of and each subset
Volldistributivität - dual is defined.
The term Volldistributivität without additive is used in different ways:
- It may mean that one is met by these two conditions and in the other case it is called dual - volldistributiv or explicitly using the above designation.
- It may mean that both conditions are met.
- It may mean that the following infinite distributive law and its dual form applies
For all three terms apply:
Each volldistributive Association is distributive and every finite distributive Association is volldistributiv.
A complete distributive lattice need not be volldistributiv, as the example shows.