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Fig. 1: COMPLEMENTED DISTRIBUTIVE t-LATTICE ON FACTORS OF 30
                           2 Ú 3 Ú 5 = 30 (max, RANK=3, FILE=1)
                                      /\
                                     /  \
                                    /  | \
                                   /   |  \
                                  /    |   \
                                 /     |    \
                                /      |     \
                               /       |      \
                              /        |       \
                             /         |        \
                        2 Ú 3 = 6 2 Ú 5 = 10  3 Ú 5 = 15 (RANK=2, FILE=3)
                            |\        / \       /|
                            | \      /   \     / |
                            |  \    /     \   /  |
                            |   \  /       \ /   |
                            |     /         \    |
                            |    / \       / \   |
                            |   /   \     /   \  |
                            |  /     \   /     \ |
                            2          3         5 (3 atoms, RANK=1. FILE=3)
                             \         |       /
                              \        |      /
                               \       |     /
                                \      |    /
                                 \     |   / 
                                  \    |  /
                                   \   | /
                                       1 (min, RANK=0, FILE=1)
Please note the FILE-RANK distribution: F=1 for R=0; F=3 for R=1; F=3 for R= 2; F=1 for R=3. This FILE distribution of 1, 3, 3, 1 is the third row of a Pascal or Binomial Table (counting top row as zeroth). This, in general, is the case: Now, the lattice-pattern resembles the hierarchical pattern in bureaucracy and military service. So I create, for this pattern, the label "bierarch", that is, b(inomial h)ierarch(y). Hence, I shall describe the t-lattice pattern by the label "bierarch". (We'll see that part of the pattern of an o-lattice is bierarch (as with t-lattice), but part of an o-lattice pattern is very different from that of the t-lattice.)

A t-lattice is (a.k.a.) a complemented distributive lattice. What does "complement" mean? Definition: elements complementary if, and only if, they join only at max and meet only at min. (That is, at "top" and "bottom" of the lattice.) Thus, the following are the complementary pairs of the above lattice: {1, 30}, {2, 15}, {3,10}, {5, 6}. Thus, in a t-lattice, every element has a complement. (This is not, in general, the case for an o-lattice.

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