> mentioned this in my previous comment

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.