Modular lattice

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Hasse diagram of N₅, the smallest non-modular lattice.

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:

Modular law: x ≤ b implies x ∨ (a ∧ b) = (x ∨ a) ∧ b.

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

Every distributive lattice is modular.

In a not necessarily modular lattice, there may still be elements b for which the modular law holds in connection with arbitrary elements a and x (≤ b). Such an element is called a modular element. Even more generally, the modular law may hold for a fixed pair (a, b). Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity.

1 Introduction
2 Diamond isomorphism theorem
3 Modular pairs and related notions
4 Footnotes
5 External links
6 References

[edit] Introduction

The modular law can be seen (and memorized) as a restricted associative law that connects the two lattice operations similarly to the way in which the associative law λ(μx) = (λμ)x for vector spaces connects multiplication in the field and scalar multiplication. The restriction x ≤ b is clearly necessary, since it follows from x ∨ (a ∧ b) = (x ∨ a) ∧ b.

Nonmodularity of N₅.

It is easy to see that x ≤ b implies x ∨ (a ∧ b) ≤ (x ∨ a) ∧ b in every lattice. Therefore the modular law can also be stated as

Modular law (variant): x ≤ b implies x ∨ (a ∧ b) ≥ (x ∨ a) ∧ b.

By substituting x with x ∧ b, the modular law can be expressed as an equation that is required to hold unconditionally, as follows:

Modular identity: (x ∧ b) ∨ (a ∧ b) = [(x ∧ b) ∨ a] ∧ b.

This shows that, using terminology from universal algebra, the modular lattices form a subvariety of the variety of lattices. Therefore all homomorphic images, sublattices and direct products of modular lattices are again modular.

The smallest non-modular lattice is the "pentagon" lattice N₅ consisting of five elements 0,1,x,a,b such that 0 < x < b < 1, 0 < a < 1, and a is not comparable to x or to b. For this lattice x ∨ (a ∧ b) = x ∨ 0 = x < b = 1 ∧ b = (x ∨ a) ∧ b holds, contradicing the modular law. Every non-modular lattice contains a copy of N₅ as a sublattice.

Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity.

[edit] Diamond isomorphism theorem

For any two elements a,b of a modular lattice, one can consider the intervals [a ∧ b, b] and [a, a ∨ b]. They are connected by order-preserving maps

φ: [a ∧ b, b] → [a, a ∨ b] and

ψ: [a, a ∨ b] → [a ∧ b, b]

that are defined by φ(x) = x ∨ a and ψ(x) = x ∧ b.

In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.

Failure of the diamond isomorphism theorem in a non-modular lattice.

The composition ψφ is an order-preserving map from the interval [a ∧ b, b] to itself which also satisfies the inequality ψ(φ(x)) = (x ∨ a) ∧ b ≥ x. The example shows that this inequality can be strict in general. In a modular lattice, however, equality holds. Since the dual of a modular lattice is again modular, φψ is also the identity on [a, a ∨ b], and therefore the two maps φ and ψ are isomorphisms between these two intervals. This result is sometimes called the diamond isomorphism theorem for modular lattices. A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.

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

[edit] Modular pairs and related notions

The centred hexagon lattice S₇, also known as D₂, is M-symmetric but not modular.

In any lattice, a modular pair is a pair (a, b) of elements such that for all x satisfying a ∧ b ≤ x ≤ b, we have (x ∨ a) ∧ b = x, i.e. if one half of the diamond isomorphism theorem holds for the pair.^[1] An element b of a lattice is called a (right) modular element if (a, b) is a modular pair for all elements a.

A lattice with the property that if (a, b) is a modular pair, then (b, a) is also a modular pair is called an M-symmetric lattice.^[2] Since a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric. In the lattice N₅ described above, the pair (b, a) is modular, but the pair (a, b) is not. Therefore N₅ is not M-symmetric. The centred hexagon lattice S₇ is M-symmetric but not modular. Since N₅ is a sublattice of S₇, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.

M-symmetry is not a self-dual notion. A dual modular pair is a pair which is modular in the dual lattice, and a lattice is called dually M-symmetric or M^*-symmetric if its dual is M-symmetric. It can be shown that a finite lattice is modular if and only if it is M-symmetric and M^*-symmetric. The same equivalence holds for infinite lattices which satisfy the ascending chain condition (or the descending chain condition).

Several less important notions are also closely related. A lattice is cross-symmetric if for every modular pair (a, b) the pair (b, a) is dually modular. Cross-symmetry implies M-symmetry but not M^*-symmetry. Therefore cross-symmetry is not equivalent to dual cross-symmetry. A lattice with a least element 0 is ⊥-symmetric if for every modular pair (a, b) satisfying a ∧ b = 0 the pair (b, a) is also modular.

[edit] Footnotes

^ The French term for modular pair is couple modulaire. A pair (a, b) is called a paire modulaire in French if both (a, b) and (b, a) are modular pairs.
^ Some authors, e.g. Fofanova (2001), refer to such lattices as semimodular lattices. Since every M-symmetric lattice is semimodular and the converse holds for lattices of finite length, this can only lead to confusion for infinite lattices.