§1C1 Four inputs. What follows is an example of why fourinput operations are not studied in this report: they are unwieldy.
For comparison, here are the 5 = 3! − 1 ways that commutativity could be defined for a threeinput operation O:
one input is fixed:
 no input is fixed:

Here are the 23 = 4! − 1 ways that commutativity could be defined for a fourinput operation N:
two inputs are fixed:
 one input is fixed:
 no input is fixed:

23 is too many formulas for most practical purposes.
§1C2 Majority operation. Although not a TQG, the simplest and most familiar nondecomposable ternary function is the majority operation M over values a and b, defined by:
Suppose that there were a decomposition M (p, q, r) = F (p, G (q, r)). Then we would have:
Observe:
§1C3 Cross product. There is a recognized ternary operation that delivers a result of the same data type as its inputs; it is a particular fourdimensional generalization of the cross product.
Let A be the vector [ a_{0}, a_{1}, a_{2}, a_{3} ]; similarly for B and C. Also let W = { W_{0}, W_{1}, W_{2}, W_{3} } be an orthonormal basis of an 4dimensional vector space. Then it becomes convenient to define this cross product as the determinant of a formal matrix. (This report uses an HTML table to display the matrix.)
A × B × C = det 

To write this another way, expand the determinant. Let R = A × B × C. Then:
r_{0} =  + a_{3}b_{2}c_{1}  − a_{2}b_{3}c_{1}  + a_{2}b_{1}c_{3}  − a_{3}b_{1}c_{2}  + a_{1}b_{3}c_{2}  − a_{1}b_{2}c_{3} 
r_{1} =  + a_{2}b_{3}c_{0}  − a_{3}b_{2}c_{0}  + a_{3}b_{0}c_{2}  − a_{2}b_{0}c_{3}  + a_{0}b_{2}c_{3}  − a_{0}b_{3}c_{2} 
r_{2} =  + a_{1}b_{0}c_{3}  − a_{0}b_{1}c_{3}  + a_{0}b_{3}c_{1}  − a_{1}b_{3}c_{0}  + a_{3}b_{1}c_{0}  − a_{3}b_{0}c_{1} 
r_{3} =  + a_{0}b_{1}c_{2}  − a_{1}b_{0}c_{2}  + a_{1}b_{2}c_{0}  − a_{0}b_{2}c_{1}  + a_{2}b_{0}c_{1}  − a_{2}b_{1}c_{0} 