I'm checking whether a multiplication table (Cayley table) satisfies the inverses axiom. Suppose I've already confirmed the identity axiom (but nothing else).
Which (if either) of the following statements are correct?
a) A multiplication table satisfies the inverses axiom if and only if the identity element appears once in each row and in each column and its occurrences are symmetrical with respect to the main diagonal.
b) A multiplication table satisfies the inverses axiom if and only if the identity element appears at least once in each row and in each column and its occurrences are symmetrical with respect to the main diagonal.
I feel that b) is correct, but some textbooks seem to suggest a).
Any help would be greatly appreciated,
Jack
If you're talking about groups then yes, the inverse axiom often states that there exists an inverse, but doesn't state that the inverse is unique. However with the other axioms the inverse can be shown to be unique. Suppose that $b$ and $c$ are inverse to $a$ then we have:
$$b = b1 = b(ac) = (ba)c = 1c = c$$
So basically they're both correct if you have associativity. Since it doesn't matter some books may also use a stronger inverse axiom that says there's a unique inverse.
(IIRC one can also weaken the axioms somewhat. You can replace the identity and inverse with same one-sided variants and still have a group that would lead to a third possibility of table description)