Lattice (order)

A lattice is in mathematics, a structure that can be completely described both as an organizing structure as well as the algebraic structure. When ordering structure an association is characterized in that there are any two elements a, b ​​a supremum, that is a ( uniquely determined ) smallest element that is greater or equal to a and b, and vice versa an infimum, a greatest element, is less than or equal to a and b. As algebraic structure, a dressing is characterized in that there are two associative and commutative operations for which the absorption characteristic laws: for arbitrary elements and.

For each occurring in the lattice theory algebraic statement there is a direct "translation " into an order statement and vice versa. This translation is also clearly understand in most cases. The ability to interpret results twice and thus better understand them, makes the investigation and the use of statements from the lattice theory so interesting.

Although this dual characterization at first glance looks very specifically, associations often occur:

• Which occur, for example, in set theory, logic and Boolean algebras are as Schaltalgebren associations.
• Total orders that occur eg in the different number ranges, such as ( natural numbers), ( integers ), ( rational numbers ) or ( real numbers ) are associations.
• For any natural number, the amount of the divider (ranked by the divisibility ) an association.
• The substructures of any algebraic or other structure form an association ( with the subset relation as an order ).

• 2.1 Neutral elements
• 2.2 Complementary elements

• 3.1 Modular Assembly
• 3.2 Distributive associations
• 3.3 Boolean algebras
• 3.4 Full associations
• 3.5 Finite Length associations
• 3.6 Compact elements and algebraic groups

• 5.1 sublattices
• 5.2 Partial associations
• 5.3 Ideals and filters

• 7.1 Total minor amounts
• 7.2 divider associations
• 7.3 subset associations
• 7.4 substructures associations of algebraic structures, sub-group associations

Clarification

Associations as algebraic structures

Associative laws:

• ;

Kommutativgesetze:

• ;

Absorption laws:

• .

For these conditions, the idempotence of both links follows:

• , and
• .

V is therefore with respect to each individual link a semilattice, ie a commutative semigroup in which every element is idempotent. The links occur in the absorption laws interact.

Associations as organizational structures

It is based on an idea of Leibniz on V is a partial order defined by:

With the absorption law to recognize the validity of the equivalences

With respect to this partial order, each two-element subset { v, w } is an upper bound ( upper bound ) and an infimum (lower limit). Here, an element s is a supremum of { v, w} if and only if

The same is true for the infimum i One can show by induction that every non-empty finite subset has a supremum and an infimum. We write general the supremum of a set M as M, and the infimum of M as M, if they exist.

Conversely, it is a partially ordered set in which each two-element subset has an infimum and a supremum, define:

• And.

The two links then meet the association axioms, as one easily checks easily.

Hasse diagrams for some examples

A finite partially ordered set (M, ≤ ) can be represented by a directed graph, called the one Hasse diagram.

If one arranges the graph so that all edges are directed " bottom-up ", then you can see the order easily:

Special elements in associations

Neutral elements

If the link has a neutral element,

Then it is uniquely determined and it is called the zero element of the association. Bzgl. is absorbent and with respect to the order the smallest element:

This is called the association then bounded from below.

If the link has a neutral element,

Then it is uniquely determined and it is called the identity element of the association. Bzgl. is absorbent and with respect to the order the largest element:

This is called the association then bounded above.

An association is called bounded if it is bounded from below and upwards, ie for both links has a neutral element.

Complementary elements

For a given element a of a limited association is called an element b with the property

• And

A complement of a

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.

But it is true: In a distributive bounded Association, the complement of an element a in the case of its existence is uniquely determined. It is often writes as ac ( especially in subset associations ), ¬ a ( mainly for applications in logic) or ā.

Applicable in any bounded Association

• ¬ 0 = 1, ¬ 1 = 0

In a distributive organization shall be limited: If a is a complement ¬ a has, then ¬ a has a complement, namely:

• ¬ (¬ A) = A.

Special associations

Modular Assembly

A lattice V is called modular if the following holds:

• For everyone.

In turn, are each equivalent to a lattice V:

• V is modular.
• For everyone.
• For everyone.
• For everyone.

A non-modular Association always contains the association as a sublattice.

Distributive lattices

A lattice V is called distributive if the links are distributive in two ways:

• For all and
• For everyone.

Since these two statements are equivalent, 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 ": an association has neither a sublattice of the form or a form, then it is distributive.

Distributive lattices are also to characterize different because Birkhoff (1933 ) and Stone ( 1936) have shown:

Boolean algebras

A distributive complementary association is called Boolean algebra or Boolean Association;

A further generalization, when the complements instead only relative Pseudokomplemente be required is Heyting algebra.

Full associations

A lattice V is called complete if every (even the empty as well as, where appropriate, infinite) subset of a supremum and an infimum has.

It is sufficient to require the existence of the supremum for each subset M, as it is

Every complete lattice V is limited to

• And

Every finite, non-empty lattice V is complete, ie it also limited.

Finite length Associations

If every respect to the ordering totally ordered subset ( chain) is finite, is called the association length finite. For many proofs within the lattice theory an association must not be finite, but it is enough if he is not very long last.

Compact elements and algebraic groups

This is called an element of a complete lattice compact ( by the related property of compact spaces in topology ) if every subset of at

Contains a finite subset, in which:

An association is called algebraic if it is complete and if every element of the supremum of compact elements.

Duality in associations

You swapped in a lattice V and the two links, one gets a new structure W. We call W the dual structure.

If you replace an arbitrary formula of the language of lattice theory and sets everywhere the two characters "" and " " mutually for each other and also replaces all "0 " to " 1" and vice versa, then it is called the resulting formula, the dual formula of.

Obviously apply in the dual to V federation W is the dual to the formulas valid in V. As in the definition of an association for each formula and the dual formula occurs, it follows that W is also a dressing, which is referred to as the Dual Association V.

From this observation follows:

• Applies a formula in all associations, then applies its dual formula in all associations.

The Modularitätsgesetz is self-dual and the two distributive laws are dual to each other and the two complementary laws are dual to each other. Therefore shall apply:

• Applies a formula in all modular or distributive in all associations or in all Boolean algebras, then we also have the dual formula of the relevant associations.

Substructures

Sublattices

A sublattice of is a subset that is associated with the restricted links of an association, that is, there are

• And for all of

Part organizations

Part of Association is a subset that is an association, that is is, a partially ordered set with supremum and infimum for finite subsets.

Of course, each sub- lattice is a sublattice, but not vice versa.

Here is one of the few places where you the difference in the approach Notes: For associations as organizational structures all part associations are substructures for associations as algebraic structures are only the sublattices substructures.

It is neither in part nor associations with sub- associations on the assumption that the neutral elements remain in the substructure. Otherwise you have to explicitly speak of a " bandage with 0 and 1"

Ideals and filters

An ideal is a sublattice of an association, the following additional condition is satisfied: are and, then. ( The definition is thus formally define the conveniences expected in a ring ).

In the order of but true. Therefore, one can also interpret the definition as follows:

Dual filters are defined to ideal: the filter is a sub- bandage, which contains together with an element, all elements of a V, which are greater than a.

Homomorphisms

Are associations and two and a function such that for all valid from

It means Verbandshomomorphismus. If, in addition bijective, then that means ( association ) isomorphism and the associations and are isomorphic.

If, complete and even

For all fulfilled, is called a complete Verbandshomomorphismus. Each full Verbandshomomorphismus is obviously also a Verbandshomomorphismus.

The class of all forms associations with these homomorphisms one category.

A Verbandshomomorphismus is also a Ordnungshomomorphismus, i.e., an isotonic illustration:

• Follows

However, not every isotonic mapping between a Verbandshomomorphismus associations.

In limited associations applies: The amount of the elements of which are represented by a Verbandshomomorphismus to the zero element of the image, form an ideal of and dual, the set of elements that are mapped to the identity element, form a filter.

Other examples of organizations

Total minor amounts

Every totally ordered set M is a distributive lattice with the links maximum and minimum. In particular, for all a, b, c of M:

• Max (a, min ( b, c) ) = min ( max ( a, b ​​) max ( a, c) ),
• Min ( A, max (b, c )) = max ( min ( A, B), min (a, c)).

Only in the case of a one-or two-element set M, the association is complementary.

Examples of the other properties:

• The closed real interval [0, 1] and the extended real line (R with ∞ and - ∞ ) are each complete distributive lattices (and thus limited ).
• The real open interval (0, 1), the amounts of R, Q and Z are each incomplete unlimited distributive associations.
• The rational interval [0, 1] Q is an incomplete limited distributive lattice.
• The quantity N0 is an incomplete distributive lattice with zero element 0

Divider associations

Looking for a natural number n the set T of all the divisors of n, then (T, gcd, lcm ) is a complete distributive lattice with unit element n ( neutral element for gcd ) and zero element 1 ( neutral element for LCM ). It is called divider Association of n The absorption and distributive laws for gcd and lcm follow this eg with the prime factorization of the properties of max and min, but they can also be derived by Teilbarkeitsbetrachtungen. The lattice is then complementary (and boolean ), if n is square-free, that is, if n has no square number as the divisor. The partial order on T is the divider ratio:

• A ≤ b if and only if a | b ( if and only if gcd (a, b) = a).

Subset associations

For a lot of the power set with the operations union and intersection with zero element forms a Boolean algebraic Association ( neutral element with respect to ) and one element ( neutral element with respect to ) as well as complement for all. It's called power sets or subset of association. The partial order on the set inclusion:

• If ( or equivalently )

(Carrier sets of) sublattices (s) of hot lot of associations ( between the associations and their means of quantities is often not distinguished). These associations are always distributive, but must be neither complete nor have neutral elements or complements. ( An example of this is the association of right -infinite real intervals of which is isomorphic to the association of real numbers. )

Substructures associations of algebraic structures, sub-group associations

For a group (G, * ) be the set A of all subgroups of G ( in general non-modular and hence not distributive ) forms an algebraic association with the links ' product of the union " and " Average ". It is called subgroup of G. Association

For example, the sub-group association between the small group of four, which just corresponds to the association, non- distributive, but modular.

Form also

• The normal subgroups of a group,
• The subgroups of an abelian ( = commutative ) group
• The sub-rings of a ring,
• The lower body of a body,
• The sub- modules of a module,
• The ideals of a ring

With analog connections a modular algebraic Association. Although the subgroups of any group and the associations of any association always give an algebraic federation, but this does not have to be modular.

In general form the ( algebraic ) sub-structures of an algebraic structure is always an algebraic Association (where the empty set is considered as a sub- structure if the set-theoretical average - ie the infimum with respect to the set inclusion - from the set of all subtrees is empty).

In particular, a lattice is then algebraically, if it is isomorphic to the association of the ( algebraic ) sub-structures of an algebraic structure ( hence the name " algebraic Association ").

{ V: V ≤ U ≤ G} If one limits the amount of sub-groups on the upper groups of a fixed subgroup U a, all these intermediate groups also form a limited association. Similarly, there are associations of intermediate rings, intermediate bodies between modules, between ideals.

Particular interest has you on the subgroup of the Galois group of a Galois association field extension L / K, because it is isomorphic to the dual interbody sublattice of L / K.