A LATTICE is a partial ordering such that each pair of elements has a least upper bound (a.k.a supremum, minmax) and a greatest lower bound (a.k.a. infimum, maxmin) -- where an ordering is partial if it is also reflexive and antisymmetric.

In the "ParOrder" hyperlink, I explained partial ordering and reflexivity, antisymmetry. Now, I'll explain the "least upper bound" and "greatest lower bound" properties which transform a partial ordering into a lattice.

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.