A complemented distributive lattice is described in several files at this Website as a t-lattice, respecting the measure of type (a.k.a. kind), but not the measure of order (a.k.a. degree). (That is, multiple tokens of a type are ignored.) It is recorded in the literature (say, Sets, Lattices, and Boolean Algebras, J. A. Abbott, and Lattice Theory, T. Donellan) that such a lattice can be decomposed in to a direct product of two-chains (elementary simple orderings), where a two-chain consists of a lattice of two elements.

Let us

• denote the "bottom" of the two-chain by 1;

• denote its "top" (Or second element) by t_i for typon of index i;

• denote the complemented distributive lattice by C;

• and denote Cartesian product by the usual "X".

Then we can formulate:

			t1	t2		tk
                        |       |       ....    |
                        |  X    |   X         X |
              	    C » |       |               |		
                        |	|		|
			1       1               1

However, although the distributive lattice -- in which not every element is complemented -- is recognized in standard lattice theory, the above decomposition will not fit it, and no attempt has been done to remedy this omission.

On discovering this, and concerned to further develop o-math beyond factor theory and distributive lattices, I set about finding a decomposition for any distributive lattice.

---

The immediate problem -- which is not adequately discussed in the literature -- is that

• a t-lattice uses only the join operation to generate the lattice, that is, all elements except the atoms (elements of rank one) are join-reducible;

• an proper o-lattice (a.k.a. distributive lattice) has at least one nonatomic element which is not join-reducible, yet no operation appears in the literature to provide it.

When I became aware of t-decomposition, I searched the literature for extension of it to the distributive lattice, but, as expected, I found none. So I set about devising such an extensio. In November, 1975, I published an Abstract in Notices of The American Mathematical Society explicating such a decomposition.

---

As shown in the file hyperlinked from this present file for the lattice on factors of twelve, the join-reducible orderings from 2, 3 to 6 are explicated in terms of a join operation (2 Ú 3). But the ordering between 2, 4 is join-irreducible, hence, must be explicated in terms of a new operator.

Similarly, the ordering from 4,6 to 12 is only partially explicated, again requiring this new operation to complete the explication. As shown in the Hasse Graph there, this new operation is labeled "aug" (from augment) and denoted @.

DEFINITION: A JOIN-IRREDUCIBLE ORDERING IS ACHIEVED BY THE OPERATION OF aug (@).

Given this new operation, it is possible to write a decomposition for an o-lattice in terms of two-chains:

1. write the requisite two-chains (example: {1, 2}, {1, 2}, {1,3});

2. form their Cartesian Product (example: {1, 2} X {1, 2} X {1,3} = {«1,1,1», «1,2,1», «2,1,1», «2,2,1», «1,1,3», «1,2,3», «2,1,3», «2,2,3»});

3. application of aug to these ordered sets involves multiplying their components (1x1x1 = 1, 1x2x1 = 2, 2x1x1 = 2, 2x2x1 = 4, 1x1x3 = 3, 1x2x3= 6, 2x1x3 = 6, 2x2x3 = 12) -- then collecting these results in a set ({1, 2, 2, 4, 3, 3, 6, 6, 12});

4. finally, reducing this t-set ({1, 2, 3, 4, 6, 12}).

In the (above) example, we have:

                 2        2       3             12
                 |        |       |             /\
                 |    @   |   @   |  =         /  \
                 |        |       |           4    6
                 1        1       1           |   / \
                                              |  /   \
                                              | /     \ 
                                              |/       \
                                              2         3
                                               \       /
                                                \     /
                                                 \   /
                                                  \ /
                                                   1 

---

Given these "tools" (further explained in "GEOMETRIC PROGRESSIONS & CHAINS", this Website), we can now state -- as an advance in ordinology -- the homologue of "The Fundamental Theorem of rithmetic".

The Fundamental Theorem of Lattice Theory: Every distributive lattice can be written as the product of two-chains (except, perhaps, as to the list-ordering of the chains).

Drawing upon the spectral language of linear vector theory, we can speak of the above as spectral decomposition of lattices.

Can you see any way to extend this?