Numbers as the Minimal Structures Prescribed by Operational Description: A Structural Selection from Operatiology and Noology

from itertools import permutations, product

elems = ["e", "A", "B"]

def mul(x, y):
    if x == "e":
        return y
    if y == "e":
        return x
    return y

table = {(x, y): mul(x, y) for x in elems for y in elems}
assert table[("A", "B")] == "B" and table[("B", "A")] == "A"
assert table[("A", "A")] == "A" and table[("B", "B")] == "B"
print("A o B = B, B o A = A, A o A = A, B o B = B : confirmed")

invertible = [x for x in elems
              if any(table[(x, y)] == "e" and table[(y, x)] == "e"
                     for y in elems)]
print("Invertible elements:", invertible)

for x in ("A", "B"):
    powers = [x]
    for _ in range(4):
        powers.append(table[(x, powers[-1])])
    print("Iterates of", x, ":", powers)

auts = []
for p in permutations(elems):
    f = dict(zip(elems, p))
    if all(f[table[(x, y)]] == table[(f[x], f[y])]
           for x in elems for y in elems):
        auts.append(f)
print("Number of automorphisms:", len(auts))
for f in auts:
    print("  ", f)
swap = [f for f in auts if f["A"] == "B"][0]
print("Swap squared is identity:",
      all(swap[swap[x]] == x for x in elems))

maps2 = list(product(range(2), repeat=2))
count = 0
for a, b in product(maps2, repeat=2):
    act = {"e": (0, 1), "A": a, "B": b}
    ok = all(tuple(act[x][act[y][i]] for i in range(2)) == act[table[(x, y)]]
             for x in elems for y in elems)
    if ok and len(set(act.values())) == 3:
        count += 1
print("Faithful actions of Q on two points:", count)

maps3 = list(product(range(3), repeat=3))
count = 0
for a, b in product(maps3, repeat=2):
    act = {"e": (0, 1, 2), "A": a, "B": b}
    ok = all(tuple(act[x][act[y][i]] for i in range(3)) == act[table[(x, y)]]
             for x in elems for y in elems)
    if ok and len(set(act.values())) == 3:
        count += 1
print("Faithful actions of Q on three points:", count)