Modular lattice

A modular organization in the sense of order theory is an association that meets the following self -dual condition ( Modularitätsgesetz ):

Modular associations occur in algebra and many other areas of mathematics. For example, are the subspaces of a vector space (and more generally the submodules of a module over a ring ) a modular lattice.

Every distributive lattice is modular.

In a non- modular lattice, it can still give elements b, together with any elements a and x satisfy (under the condition x ≤ b). Modularitätsgesetz the Such an element b is called modular element. More generally, one pairs (a, b ) consider elements that meet the Modularitätsgesetz for all elements x. Such a pair is called a modular pair, and there are several with the Semimodularität related generalizations of modularity that are based on this concept.

Introduction

The Modularitätsgesetz may be considered as a restricted associative law, which links the two lattice operations in a manner similar to the associative law λ ( μx ) = ( λμ ) x for vector spaces the body multiplied by the scalar multiplication. The restriction x ≤ b is necessary because it follows from x ∨ (a ∧ b) = (x ∨ a) ∧ b.

One can easily check that for x ≤ b in each dressing x ∨ (a ∧ b ) ≤ (x ∨ a) ∧ b follows. Therefore, one can the Modularitätsgesetz also be formulated as follows:

By the term x ∧ b used for x, one can express the Modularitätsgesetz as follows by an equation that must be met without preconditions:

This shows (using terms from the universal algebra) that the modular associations form a subvariety of the variety of associations. Therefore, all homomorphic images, sublattices and direct products of modular organizations are modular again.

The smallest non-modular organization is the " federation Pentagon " N5, which consists of five elements 0.1, X, A, B, such that 0

According to Richard Dedekind, who discovered the Modularitätsgesetz, modular assemblies are sometimes still referred to today as Dedekind associations.

Diamond Isomorphism Theorem

For any two elements a, b of a modular lattice can be the intervals [a ∧ b, b ] and [ a, a ∨ b] consider. Between them there is the order -preserving mappings

Defined by φ (x ) = x ∨ a and ψ (x ) = x ∧ b.

Counter-example to the diamond isomorphism theorem in a non- modular lattice

The composition ψφ is an order-preserving mapping of the interval [a ∧ b, b] into itself, which is also the inequality ψ ( φ (x)) = ( x ∨ a) ∧ b ≥ x met. The example shows that this inequality I.A. need not be an equation. In a modular lattice, however, always applies the equation. Since the association to a dual modular organization is dual again, φψ as the identity mapping to [ A, A ∨ B ]; therefore φ and ψ are isomorphisms between these two intervals. This result is sometimes referred to as diamond isomorphism theorem for modular assemblies. A lattice is modular if and only if the diamond isomorphism theorem is true for every pair of elements.

The diamond isomorphism theorem for modular organizations is analogous to the third isomorphism theorem in algebra, and it is a generalization of the association set.

Modular couples

In each association is meant by a modular pair of a pair (a, b ) of elements such that for all elements x, the a ∧ b ≤ x ≤ b satisfy the equation (x ∨ a) ∧ b = x. In other words, the modular pairs are the pairs for which valid one half of the diamond Isomorphiesatzes. The French term for " modular pair " is modulaire couple. A pair (a, b ) is called in French paire modulaire if both (a, b ) and ( b, a) modular pairs. An association element b is called (right ) modular element if for all elements a the pair ( a, b) is modular.

Some associations the property that for each modular pair (a, b ) is the pair ( b, a) modular. Such an association is called M- symmetric association. Some authors, for example Fofanova, designate such organizations as semi-modular organizations. Since each M- symmetric association is semi- modular and for associations of finite length the converse holds, this can only be for certain infinite associations lead to confusion. Since a lattice is modular then, if every pair of elements is modular, each modular bandage is M- symmetric. In the above described association N5 the pair ( b, a) is modular, and not the pair (a, b). Consequently, N5 not M- symmetric. The provided with a center Hexagon Association S7 is M- symmetric, but not modular. Since N5 is a sublattice of S7, the M -symmetric associations do not form a subvariety of the variety of associations.

M- symmetry is not self-dual concept. A dual- modular pair is a pair that is in the dual modular association and an association is dual M- symmetrical or symmetrical if the M * M- dual dressing is symmetrical. It can be shown that a finite lattice is then modular, if it is symmetrical, and M- M * - symmetrical. The same equivalence is true for infinite associations fulfilling the ascending chain condition ( the descending chain condition or ).

Some less important terms are closely related at the moment. An association is called cross- symmetric, if for each modular pair (a, b ) is the pair ( b, a) dual modular. From cross symmetry follows M- symmetry, but not M * symmetry. Therefore, cross symmetry is not binary cross symmetry equivalent. An association with a smallest element 0 is symmetric ⊥ if for each modular pair (A, B ) which satisfies A ∧ B = 0, the pair ( B, A ) is also modular.