Let a + a + a = a. Adding the inverse of a to both sides, we get a + a = 0.
Let a + a = 0. Adding a to both sides, we get a + a + a = a.
> I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation"
Yes. As I mentioned in my previous comment, "In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically."
In multiplicative notation, the statement becomes: a·a·a = a holds if and only if a·a = e, where e denotes the identity element.
You did. I'm sorry I glossed over the ending to your comment. I was focused on a counterexample I was working on and went only on my memory of group theory.
> Adding the additive inverse of a, i.e., -a from both sides, we get a + a = 0.
That assumes associativity, but that's a nitpick, not a real objection.
In reality, I got a bit tired and mentally shifted the question to a + a + a = 0, not a + a + a = a. That of course has numerous examples. But is irrelevant.
Thanks for taking the time for the thoughtful, and non-snarky, response. Sorry if I was abrupt before.
> That assumes associativity, but that's a nitpick, not a real objection.
I don't think that is a valid nitpick. My earlier comments assume associativity because a group operation is associative by definition. If we do not allow associativity, then the algebraic structure we are working with is no longer a group at all. It would just be a loop (which is a quasigroup which in turn is magma).
> Thanks for taking the time for the thoughtful, and non-snarky, response. Sorry if I was abrupt before.
No worries at all. I'm glad to have a place on the Internet where I can talk about these things now and then. Thank you for engaging in the discussion.
Both "if" and "only if" are correct.
Let a + a + a = a. Adding the inverse of a to both sides, we get a + a = 0.
Let a + a = 0. Adding a to both sides, we get a + a + a = a.
> I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation"
Yes. As I mentioned in my previous comment, "In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically."
In multiplicative notation, the statement becomes: a·a·a = a holds if and only if a·a = e, where e denotes the identity element.