Go from the proto-educable (partial ordering) term, SUBORDINATE, to its formal equivalent, SUBORDERING. This suggests, for any term of the SUBORDERING a "lower" and "upper", and this suggests (for FINITE ORDERING) a MINIMUM and a MAXIMUM. For the LATTICE, I'll call these, respectively, "MAX" and "MIN". (Unfortunately, the standard terms are, respectively, "0" and "1", which confused the laity. And it conflicts with my use of "indicator tables" for lattices -- similar to those used in SET-THEORY -- with indicators of "0" and "1".)

But for any term t of a SUBORDERING, a "lower" term of t is a "minimum" for t. So t may have many "minima". But, for a FINITE ORDERING, there must be a maximal minimum or a MAXIMIN. (The REFLEXIVE PROPERTY of a PARTIAL ORDERING allows a term to be its own minimum).

I declare that ANY TWO TERMS OF THE LATTICE, THERE EXISTS A MAXIMIN FOR THEM.

Similarly, a term t has a "higher" term which is a "maximum" of it. But for a FINITE ORDERING, there must be a minimal maximum, or MINIMAX. (RFELEXIVITY allows a term to be its own maximum.)

I declare that, FOR ANY TWO TERMS OF THE LATTICE, THERE EXISTS A MINIMAX. (RFLEXIVITY and ANTISYMMETRY in a PARTIAL ORDERING allows this.) And I declare that, FOR ANY TWO TERMS OF THE LATTICE, THERE EXISTS A MAXIMIN. (REFLEXIVITY and ANTISYMMETRY allows this.) these two PROPERTIES TRANSFORM A PARTIAL ORDERING INTO A PARTIAL ORDERING INTO A LATTICE.

But, for our purposes, LATTICES separate into FOUR TYPES (with the last two primary):

1. NONMODULAR LATTICES;
2. MODULAR LATTICES;
3. DISTRIBUTIVE LATTICES;
4. COMPLEMENTED DISTRIBUTIVE LATTICES
.

The first two types can be explained by the notion of SUBLATTICE: A SUBSYSTEM OF A LATTICE, WHICH ALSO QUALIFIES AS A LATTICE.

Any LATTICE containing the following "5-point lattice" as a SUBLATTICE is of TYPE 1, above -- failing in MODULARITY, DISTRIBUTIVITY, COMPLEMENTATION:

               *
              / \     NONMODULAR (fails MODULARITY)
             /   \
             *   *    (cannot assign RANK "metric", as
             |   /     it appears in LATTICE below)
             *  /
             | /     
             |/       
             * 

Any LATTICE containing the following "5-point MODULAR lattice" as a SUBLATTICE is of TYPE 2, above -- failing DISTRIBUTIVITY:

        Rank2    *  MAX
                /|\      MODULAR but NONDISTRIBUTIVE
               / | \
        Rank1 *  *  *   (MODULAR and DISTRIBUTIVE LAWS
               \ | /     explained below)
                \|/
        Rank0    *  MIN  ("quantum logic" fits MODULAR
                    LAW due to COMBINATORICS)    

The DISTRIBUTIVE LAW of LATTICES resembles the DISTRIBUTIVE LAW OF ARITHMETIC -- a X (b + c) = a X b + a X c -- except THAT THERE ARE TWO DISTRIBUTIVE LAWS IN LATTICE THEORY. Let denote JOIN; and denote MEET. Then, the LATTICE DISTRIBUTIVE LAWS, for LATTICE terms, A, C, D, are:

• A (C D) = (A C) (A D);
• A (C D) = (A C) (A D)
.

MODULAR LAW: For terms A, C, D, where D A (D dominates A), with join, meet as ebfore,

• A (C D) = (A C) D.

In a COMPLEMENTED DISTRIBUTIVE LATTICE, EVERY non-MAX and non-MIN TERM HAS A COMPLEMENT: THEY JOIN ONLY AT MAX, AND MEET ONLY AT min. Elsewhere, I've referred to a COMPLEMENTED DISTRIBUTIVE LATTICE as a "t-lattice". (In my mathtivity, "Pecking Order", a PLAIN PECKING ORDER is an example.) And I've referred to a NONCOMPLEMENTED DISTRIBUTIVE LATTICE as an "o-lattice". (A MIXED PECKING ORDER is an example.)

An example of a t-LATTICE is THE FACTOR LATTICE of a t-NUMBER -- a number such as 30 = 2 X 3 X 5 CONTAINING EACH PRIME GNOMON EXACTLY ONCE. An example of a o-LATTICE is THE FACTOR LATTICE of an o-NUMBER -- a number such as 12 = 2 X 2 X 3, containing SOME PRIME (here, 2) GNOMON MULTIPLY.

To explain why COMPLEMENTATION fails (as in proper TYPE 4 LATTICES), I need to explain NONREDUCIBILITY. Note that, in the MODULAR LATTICE, MAX is the JOIN of 3 LATTICE terms. It is said to the JOIN-REDUCIBLE. The terms of RANK1 have a MEET (MAXIMIN) in the MIN of the LATTICE (RANK0). The RANK1 terms are said to be JOIN-IRREDUCIBLE. In fact, All LATTICE terms of RANK 1 are JOIN-IRREDUCIBLE, hence, trivially so. Terms of RANK 1 are called "atoms" of the LATTICE. (Unfortunate term, but standard. I didn't did it!) If any NONATOMIC TERMS (non-Rank-1 terms) are JOIN-IRREDUCIBLE, they are PROPERLY JOIN-IRREDUCIBLE. A DISTRIBUTIVE LATTICE WITH ONE OR MORE PROPER JOIN-IRREDUCIBLES FAILS COMPLEMENTATION.

So the last two TYPES are primary to us.