Modular lattices and subalgebras

Modular lattices can be considered from two perspectives: (1) the general order-theoretic perspective which examines the class of modular lattices and (2) the properties of its members and the specific modular lattices that arise in various other branches of mathematics. We will now take the later approach. Here are some of the types of modular lattices that arise in most of the branches of abstract algebra:

• Lattices of normal subgroups
• Lattices of submodules
• Lattices of ideals of a ring
• Lattices of normal intermediate field extensions of galois extensions

In general, if we have an algebraic structure that has a kernel associated with it, and associated special subalgebras that define all quotients then in all the standard cases (groups, rings, modules, etc) these subalgebras form a modular lattice as a subalgebra system. That modules form a modular lattice was proven by Dedekind, and it is not hard to see that ideals are a special case of submodules. The last case is simply a corollary of the fundamental theorem of galois theory.

Proposition. the lattice of normal intermediate field extensions of a galois extension form a modular lattice

Proof. the lattice of normal subgroups of the galois group of the field extension is modular. By the fundamental theorem of galois theory (FTGT), we know that the lattice of normal intermediate field extensions of a galois extension is order-dual to the lattice of normal subgroups. But since modularity is a self-dual condition, the lattice of normal intermediate field extensions is modular.

Subalgebra-modular structures:
Supposing that we have an algebraic structure like a ring then we can form from these structures subalgebra systems (suborders of Sub(A)) that are modular. A separate question is under what conditions is Sub(A) modular? We can now answer this to some extent using the concepts already covered. Here are algebraic structures for which Sub(A) is modular:

• Dedekind groups
• Commutative groups (a special case of dedekind groups)
• Modules
• Fields which are galois-dedekind over their prime subfield
• The ring of integers Z (all subrings are ideals)

Commutative groups can be seen to be subalgebra-modular in two different ways: (1) by the fact that they are dedekind or (2) by the fact that commutative groups can be represented as (not necessarily faithful) Z-modules. That commutative groups are subalgebra-modular has implications in group theory, because it means from the commuting graph we can infer not only subalgebras but also properties of the lattice structure. Dedekind groups are not the only subalgebra-modular groups and in general subalgebra-modular groups are refered to as iwasawa groups.

Complements:
Most of these subalgebras are related normal subgroups in some way, but normal subgroups have another important property. In the lattice of normal subgroups, complementary members characterize direct product representations of the group. It is not hard to see then, how complements are unordered, as a given subgroup which characterizes the group by direct product with some other complementary subgroup is not going to produce the same direct product with a subalgebra of that other complementary group. Therefore, the property that complements in a modular lattice are unordered (which was proven entirely using order theory) is a pre-requisite for the complements to determine direct products.