• the occupancy/nonoccupancy of set elements in set theory or probability theory;
• truth/falsity of statements of a statement logic;
• the partial ordering in a lattice (perhaps only so used here).

• u = f(x,y), v = g(x,y);
• du = f₁dx + f₂dy; dv = g₁dx + g₂dy;
• dudv = (f₁dx + f₂dy)(g₁dx + g₂dy) = (f₁g₂ — g₁f₁)dxdy = [f, g]_1,2dxdy -- because dxdx = dydy = 0, dydx = ¯dxdy UNDER EXTERIOR PRODUCT (OUTER PRODUCT).

• Given a scalar triangle (PL) transformed into a vector triangle (PL), the corresponding transformation of the law of cosines (usually derived from inner product) results in the inner product of vectors.
• Given two independent vectors (with null inner product) and a third independent but unspecified structure, algebraic elimination leads to two equations in three unknowns, which can be resolved by specifying a relating structure, and two choices arise for specifying that third independent structure: a 1-vector choice leads to the vector or cross product of Gibbs-Heavide vector theory; a bivector choice leads to outer product of multivectors.
• The derivation of multiproduct proceeds in these stages:
  1. a trinomial (as polynomial correspondent of a 3-D vector) is studied and found to separate into a heterogeneous subpolynomial plus a homogenous subpolynomial;
  2. the outerproduct is found correspondent to a heterogeneous polynomial, and the innerproduct to a homogeneous polynomial;
  3. a new operation is designed to conserve the heterogeneity/homogeneity of a polynomial;
  4. the application of this operation to two independent polynomials results in a sum of a heterogeneous subpolynomial and a homogeneous subpolynomial;
  5. the previous result can then be identified with the axiomatically declared multiproduct as sum of outerproduct and innerproduct, thus explaining this powerful operation (as it is not explained in the literature.

q sup {x| p q = p x}

q inf {x|p q = p x}

q sup {x| p q = x}

q sup {x| p q = p x}

\

O {,, ,

p q p ¬p

sup, inf, ,

v dt = td

K {d, t, v, i}

			SJ<--d--->JS
                          ^ \   /^
                          | v\ / |
                         t|   \  |t
                          |  / \ |
                          | /v  \|
                          V      V
                        SR<--d--->RS

                                |d  t  v  i
                               d|i  v  t  d
                               t|v  i  d  t
                               v|t  d  i  v
                               i|d  t  v  i

SR {G, }

RS {G, }

SJ {G, }

JS {G, }

A {SR, RS. SJ, JS}

. Problem: Can this be generalized to a group-theoretic relation between symtactic and semantic structures? between mathematics and metamathematics? between logic and metalogic?.