I realize this is rather an arbitrary question, but it's important to me, that I understand it and get it right, and I'm not finding the answer anywhere else.
I'm working through "A Book of Abstract Algebra" by Pinter and one problem set defines binary words and defines addition on them to be
If a and b are both of length n then $(a_1, a_2,..., a_n) + (b_1, b_2,..., b_n) = (a_1 + b_1, a_2 + b_2,..., a_n + b_n)$
, so you add them by corresponding digits. The way you add each digit is as follows:
$0 + 0 = 0$, $1 + 1 = 0$, $0 + 1 = 1$, $1 + 0 = 1$
There's more info the book gives about this but I think this is all that's pertinent to answer the question, but if not please correct me.
The question I'm trying to answer is what is the identity element of the operation of word addition, +, on the set of all binary words of length n, $\Bbb B^n$ ?
The answer is obviously a string of zeros but how do I properly denote this? $0_n$, or just 0, or a such that every $a_*i*$ is 0, or what?
I have the same question for the inverse. The inverse is obviously whatever the element in the set $\Bbb B_n$ is itself but how do you denote that?? Just a or $a_n$, etc.?
Generally, I see something like the following:
The important thing here is not the $0^n$, which is an arbitrary choice of notation (and could be any of the other choices you gave, or still others), but the exposition, which explains, more or less clearly, what the identity element actually is, so that we needn't rely on the notation for everything. Thereafter, of course, the notation can be used as a kind of shorthand for the explanation, but the first time, the explanation is important.
As André Nicolas pointed out in the comments, the inverse of $a$ is $a$ itself.