MODULAR LATTICES MODEL "POINTED SETS" (SETS WITH "FAVORED ELEMENTS")

A distributive lattice satisfied the following distributive laws, for any x,y,z of the lattice:

x Ù (y Ú z) = (x Ù y) x Ú (x Ú z);
x Ú (y Ù z) = (x Ú y) x Ù (x Ù z).
(Actually, if the lattice satisfies the first condition, it satisfies the second.)
However, it is generally known that a lattice is not distributive if it contains as a sublattice either of the 5-point lattices displayed below:
MODULAR NONMODULAR * * /|\ / \ / | \ / \ / | \ / * / | \ / | * * * / | \ | / * * \ | / \ / \ | / \ / \|/ \ / * \/ *
The left lattice (above) satisfies the modular law, a generalization of the distributive law. For any x,y,z of the lattice such that z x,
x (y z) = (x y) z.
What good is modularity? A modular lattice models the lattice of subgroups of a given group. Extremely important, since, in mathematics, a system is "nicely tied up" if it forms "a group". (Elsewhere I explain this concept to kids via "The Creeping Baby Group".) In mathematical physics, a group is indicator of a conservation law. Nuff said?
Definition: Given a set, S, and a binary operation, o, the set forms a group if, and only if

the set is closed under this operation;
for every element x in the set, there is an inverse element, x^-1 such that x o x^-1 = x^-1 o x = I, where I is the identity element of the group such that x o I = I o x = x.
Now, just as it is often desirable to determine subsets of a given set, so is it also sometimes desirable to determine subgroups of a given group. (For example, every finite group is isomorphic to a subgroup of a permutation group. So, if you understand permutation groups, you understand all finite groups.)
We have also seen that the powerset of a set, that is, the collection of all its subsets, forms a complemented distributive lattice. However, the collection of all subgroups of a given group does not form a distributive lattice, but only a modular lattice!
Why? The difference between the subset and subgroup cases consists in the following:

a set has no "favored" element such that it must appear in every subset, hence, all possibilities are allowed;
a group has one or more "favored elements" which must appear in a subgroup:

each subgroup must contain the identity, I, of the group;
if the subgroup contains an element a of the group, then this subgroup must also contain its inverse, a^-1;
thus, most of the subsets of the group are ruled out as subgroups;
so, those qualifying subsets form, not a distributive lattice, but a modular lattice.
. This is easily shown in the lattice of subgroups of D₃, the six symmetry transformations on an equilateral triangle.
We can show this by its Group Table:
CAYLEY TABLE OF D₃
o I R₁ R₂ M₁ M₂ M₃

I I R₁ R₂ M₁ M₂ M₃

R₁ R₁ R₂ I M₂ M₃ M₁

R₂ R₂ I R₁ M₃ M₁ M₂

M₁ M₁ M₃ M₂ I R₂ R₁

M₂ M₂ M₁ M₃ R₁ I R₂

M₃ M₃ M₂ M₁ R₂ R₁ I

This group is said to be of order 6, since it has 6 members. Let's look at its subgroups:

from the Table, it is clear that the identity I is a subgroup all by itself {I}, of order 1;
from the Table, we see that M₁ is its own inverse -- that is, M₁ o M₁ = I -- hence, {I, M₁} constitutes a subgroup of this group, of order 2, and the same for the other two medians, with {I, M₂} forming a subgroup of this group, also of order 2, and {I, M₃} forming a subgroup of this group, also of order 2;
from the Table, we see that R₁, R₂ are mutually inversive -- that is, R₁ o R₂ = R₂ o R₁ = I -- so {I, R₁, R₂} is a subgroup of this group, of order 3;
however, we see, from the Table, that putting any median into a 2-group involving a different median does not result in closure, which means that there is only the one subgroup of order 3;
and putting a median in with the rotation subgroup results in closure being found with all the elements of the group, so there is no subgroup of order 4, none of order 5;
trivially, the group is a subgroup of itself, of order 6;
and this is all.
So we have subgroups of order 1, 2, 3, 6. This agrees with a well known result in group theory:
Theorem (Lagrange): Given a group of order n, the orders of its subgroups are factors of n.
In Summary, we have 6 subgroups: {I}, {I, M₁}, {I, M₂}, {I, M₃}, {I, R₁, R₂}, {I, R₁, R₂, M₁, M₂, M₃}.
What happened? Given a set of 6 members, we have 2⁶ subsets -- but only 6 subgroups. This is in keeping with what, elsewhere, we describe as the antitonic process, whereby -- as structure increases, scope decreases. A set has simple structure, so broad scope; a group has extensive structure, so limited scope.
Let's look at the lattice of subgroups of D₃:
{I , R₁ , R₂ , M₁ , M₂ , M₃} | | | | | | | | | | | | | | | | {I,M₁} {I,M₂} {I,M₃} {I, R₁, R₂} \ \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ / / ------------{I}------------------
Do you see the modularity of this lattice? THE LATTICE OF SUBGROUPS OF ANY GIVEN GROUP IS MODULAR -- VIOLATING DISTRIBUTIVITY. BECAUSE THE GROUP STRUCTURE HAS FAVORED ELEMENTS:

every subgroup must contain the identity element, I;
if a subgroup contains any element, it must also contain its inverse.
(Note: For complicated historical reasons, a set with one or more favored elements is called a "pointed set:.)
Interesting. But so what?
In the file, "TOPOLOGICAL SORTING BY XX", this Website, I note that the literature records a metric in lattices known as Rank. And I note that this inspired me to create another metric known as "File" to distinguish different elements of the same Rank. Thus, we gave a coordinate system on lattices of the form [R_i, F_j]. Thus, we can rewrite the above Hasse diagrams of a modular and nomodular lattice as:
MODULAR NONMODULAR max [2,1] max [3,1] /|\ / \ / | \ / \ / | \ / [2,1] / | \ / | [1,1] [1,2] [1,3] / | \ | / [1,1] [1,2] \ | / \ / \ | / \ / \|/ \ / min [0,1] \/ min [0,1]
By the methods described in "TOPOLOGICAL ...", the modular parordering can be transformed into the simordering: [ [0,1], [1,1], [1,2], [1,3], [2,1] ]. Similarly, for the nonmodular lattice: [ [0,1], [1,1], [1,2], [2,1], [3,1] ]. (Please notice the difference between these two different simorderings.) Furthermore, the above ordered sets can be interpreted as relational descriptions of the given structures.
And in the file "TOPOLOGICAL ...", I show how to write an o-lattice (a.k.a distributive lattice) as an augment of 2-chains, homologous to writing numbers as products of primes. We can do something similar for the above nondistributive lattices.
For the modular lattice, we need to speak of 3-chains; and we need a different operator, chain-product, denoted Ó. Then, we have:
MODULAR max max max max [2,1] | | | /|\ | | | / | \ | Ó | Ó | / | \ | | | / | \ [1,1] [1,2] [1,3] = [1,1] [1,2] [1,3] | | | \ | / | | | \ | / | | | \ | / | | | \|/ min min min min [0,1]
The modular lattice as the chain=product of three 3-chains.
Similarly, for the nonmodular lattice:
NONMODULAR max max max [3,1] | | / \ | | / \ | [2,1] / [2,1] | Ó | / | | | / | [1,1] [1,2] [1,1] [1,2] | | \ / | | \ / | | \ / | | \/ min min min [0,1]
The nonmodular lattice as the chain-product of a 2-chain and a 3-chain.
In a similar way, we can decompose the subgroup lattice above by writing G º {I,R₁,R₂,M₁,M₂,M₃}. Then we have:
G G G G G ------------G------------- | | | | | | / | \ | | | | | | | / | \ | | | | | | | / | \ | | | | | | | / | \ | R1 R2 M1 M2 M3 = R1 R2 M1 M2 M3 | Ó | Ó | Ó | | \ | | | | | | | | | \ | | | | | | | | | \ \ | / | | | | | | \ \ | / | | | | | | \ \ | / | | | | | | \ \| / | I I I I I \_____I____________|
The subgroup lattice as the chain-product of four 3-chains.
In, "TRANSFORMATIONS BETWEEN ORDEXES", this Website, I show how a distributive lattice transforms into a properly modular lattice, and vice versa; and how a modular lattice transforms into a properly nonmodular lattice, and vice versa.

o	I	R₁	R₂	M₁	M₂	M₃
I	I	R₁	R₂	M₁	M₂	M₃
R₁	R₁	R₂	I	M₂	M₃	M₁
R₂	R₂	I	R₁	M₃	M₁	M₂
M₁	M₁	M₃	M₂	I	R₂	R₁
M₂	M₂	M₁	M₃	R₁	I	R₂
M₃	M₃	M₂	M₁	R₂	R₁	I