Properties [of operators from axioms]

There are k^{k2} 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 it has k=1. Of the 274,499 axiom systems of the form {… == a} where … involves ∘ up to 6 times, 32,004 allow only operators {6,9}, while 964 allow only {1,7}. The only cases of 2 or less operators that appear with k=2 are {{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}}. (See page 1174.)