Why are modular lattices important?

Question

A lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. (Join and meet.)

The ideals of a ring form a modular lattice. So do submodules of a module. These facts are easy to prove, but I have never seen any striking examples of their utility. Actually, in a seminar I took part in, the speaker said the modularity condition wasn't very natural and that there was an ongoing search for better ones (this was in the context of the Gabriel dimension and its generalization to lattices -- unfortunately, I didn't understand much of that).

I would like to see some motivation for this notion. That is, I would like to know when it is useful, and if it is natural. At the moment, it doesn't look any more natural to me than any random condition in the language of lattices. If you could shed some light on the opinion I quote in the previous paragraph, it would be very helpful as well. I would be especially interested in algebraic motivation, as I know very little about other areas if mathematics.

Answer 1

The definition of modularity looks more natural to me if I think of it as follows (rather than as a modified associativity or a weakened distributivity). Given any element $a$ of a lattice $L$, there is a rather obvious way to map any element $x\in L$ to a "nearest" element $\geq a$, namely send $x$ to $a\lor x$. Think of this map as "projecting " elements into the part of $L$ above $a$. There is, of course, a dual notion of projecting elements into the part of $L$ below a given element $b$, namely $x\mapsto b\land x$. If $a\leq b$, then we can combine these ideas to "project" any $x$ into the interval $[a,b]=\{z\in L:a\leq z\leq b\}$, namely first project $x$ above $a$ and then project the result below $b$. (One needs to check that the second projection doesn't ruin what the first achieved; the final result is still above $a$ as well as below $b$.) Again, duality provides another way to project $x$ into $[a,b]$, namely first project it below $b$ and then project the result above $a$. So in general we have two competing, equally natural notions of projecting an element $x$ into an interval $[a,b]$. Modularity eliminates the competition; it says precisely that these two projections always agree.

Answer 2

For reference:

Modularity: $x\leq b \implies x\vee(a\wedge b)=(x\vee a)\wedge b$

Algebraically it is a relaxed distributivity condition for the meet and join operations. Graphically, it means that the forbidden Pentagon diagram will have one or more of its sides crushed. I can't think of any more proof for their naturality other than the fact the submodules of a module and the set of normal subgroups of a group are all modular lattices. Groups and modules are very natural!

Similarly its cousin the distributive lattice is algebraically distributivity of meet and join over each other. Modules do not normally have a distributive lattice of submodules.

Distributive lattices are natural because their prototype is the lattice of subsets of a given set with intersection and union operations. It is known that every distributive lattice is lattice isomorphic to such a set lattice.

P.S.: I didn't know this before, but I found that von Neumann apparently made use of complemented modular lattices in his book Continuous Geometry, so I would also look there for inspiration.

Added: You were requesting some places where modularity was explicitly used. When Ward and Dilworth went about abstracting the study of ideals in a ring to "multiplicative lattices", they managed to do primary decomposition in what they called Noether lattices. These were of course supposed to generalize the lattice of ideals of a Noetherian ring, and general multiplicative lattices are far too wild, so they needed to make some additional natural requirements for the multiplicative lattice. Among these assumptions were the ACC (to make it Noetherian), the property that every element should be a join of principal elements, and finally modularity of the lattice. I'm not an expert in the topic but I think modularity was probably crucial in their proofs using residuals.

Answer 3

I'd like to complete Andreas's answer by the following consequence of the modularity property:

There is a natural isomorphism between $[a,a\vee b]$ and $[a\wedge b,b]$. In group or module theory, this is the natural isomorphism $(A+B)/A=B/(A\cap B)$. Any argument that uses this (2nd? 3rd? 1st?) isomorphism theorem applies as well to the lattice of submodules, normal subgroups, etc., and more generally any modular lattice.

Answer 4

Let $H$ be a Hilbert space, write $L(H)$ for the poset of closed subspaces of $H$. Note that $L(H)$ is a lattice: the meet is the intesection of subspaces, the join is the closure of the sum of subspaces.

Then $H$ is finite-dimensional if and only if $L(H)$ is modular.

Answer 5

I'm doing some work with modular lattices, and it seems that one situation in which they are ubiquitous is the lattice of subobjects of an object. For instance, consider the lattice of subspaces of a vector space. In general, the meet of two subobjects will be their set-intersection, which will be closed under any operation. But the join is usually bigger than the set-union, more commonly a suitable "closure" of the set-union.

E.g. the join of two sub-vector-spaces is the set of linear combinations of elements of the union of the subspaces. The common pattern is that these lattices of subobjects are modular but not distributive. A sign of this is the "modular rank law", $\rho(x) + \rho(y) = \rho(x \vee y) + \rho(x \wedge y)$.

But beware this is not universal. For instance, while linear subspaces form a modular lattice, affine subspaces do not. (The plane which is the join of a line and a point not on the line is not just the linear combinations of the point and points on the line, as that does not generate the line through the point parallel to the given line. But also, two parallel lines do not obey the modular rank identity.)

ETA: It turns out that this simplified join process is characteristic of linear lattices, which is a stronger property than modular. But most quoted examples of modular lattices are also linear.