Properties [of operators from axioms]
There are kk2
✖
\!\(\*SuperscriptBox[\(k\),\(\!\(\*SuperscriptBox[\(k\),\(2\)]\)\)]\) possible forms for binary operators with k possible values for each argument. There is always at least some operator that satisfies the constraints of any given axiom system—though in a case like a b
✖
a b it has k = 1
✖
k = 1. Of the 274,499 axiom systems of the form {… a}
✖
{… a} where … involves ∘ up to 6 times, 32,004 allow only operators {6,9}
✖
{6,9}, while 964 allow only {1,7}
✖
{1,7}. The only cases of 2 or less operators that appear with k = 2
✖
k = 2 are {{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}}
✖
{{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}}. (See page 1174.)
