In the file, "ORDINATED INDICATOR TABLES", at this Website, I note that I must be unique in assigning indicator tables ("truth tables") to factors of number -- otherwise some one would have noted in the literature that the standard tables can be assigned only to "square-free" numbers, that is, numbers have each prime factor only once. This eliminates "most" of factor arithmetic, whereas yy o-tables encompass all of factor arithmetic.

Similarly, I must be unique in assigning these tables to lattices. I reason thus because of a customary abuse of language in talking about lattices. The "top" of the lattice is labeled its "1"; its bottom, its "0". So, when one brings in indicator tables with "1" denoting "yes", "0" denoting "no" -- as is done in tables for sets and for probabilities -- then confusion takes over. Does "1" refer to the "lattice top" or to an assignment in the indicator table evaluating the entire lattice?

I resolved this conundrum by referring to the lattice "top" as "max"; to its bottom as "min". That I did not, thereupon, use "Max" and "Min" leads into the matter of "QUASI-FINITY OF LATTICES" in my title: "You can't get there from here!" (And, in another file at this Website, we see that this leads to "QUASI-FINITY OF LOGIC".)

---

Consider the factor lattice on 30 = 2 * 3 * 5:

                                      30
                                      /\
                                     /  \
                                    /  | \
                                   /   |  \
                                  /    |   \
                                 /     |    \
                                /      |     \
                               /       |      \
                              /        |       \
                             /         |        \
                            6         10        15
                            |\        / \       /|
                            | \      /   \     / |
                            |  \    /     \   /  |
                            |   \  /       \ /   |
                            |     /         \    |
                            |    / \       / \   |
                            |   /   \     /   \  |
                            |  /     \   /     \ |
                            2          3         5
                             \         |       /
                              \        |      /
                               \       |     /
                                \      |    /
                                 \     |   / 
                                  \    |  /
                                   \   | /
                                       1 

Now, let's look at the associated indicator table:

t-TABLE OF 30 = 2 * 3 * 5

1	2	3	5	6	10	15	30
0	0	0	0	0	0	0	0
0	0	0	1	0	1	1	1
0	0	1	0	1	0	1	1
0	0	1	1	1	1	1	1
0	1	0	0	1	1	0	1
0	1	0	1	1	1	1	1
0	1	1	0	1	1	1	1
1	1	1	1	1	1	1	1

We note the following about this t-Table:

• The lowest element of the lattice (factor 1) has a subtable (column) of not all zeros, hence, cannot be a MIN, but only a min.
• The highest element of the lattice (factor 30) has a subtable (column) of not all ones, hence, cannot be a MAX, but only a max.
• This lattice is homologous to all lattices derived from 3 "atoms" (elements of lattice rank one), so all these lattices have the same min, max problem.
• The same kind of problem will occur with any lattice derived from n "atoms", so all finite lattices have the same min, max problem.
• But the MAX of lattice theory should have a subtable consisting of all ones, since it is homologous to the "final table" for a valid argument in logic, as seen in the file, "HISTORY OF INDICATOR TABLES", at this Website, regarding MP; similarly, for the MIN of lattice theory, which should be homologous to a table for a logical contradiction, all zeros.
• Thus, there is a discrepancy between "theory" and "practice" which can be resolved only by a transfinitary argument, so you see "the real face of lattice theory, warts and all".

Hence, the quasi-finity of my title: "You can't get there from here!"