Usage of "commute"/"commutative" I have a very elementary terminology question:
When you say "$x$ and $y$ commute under some operation", can you only say that if the operation is generally commutative, or is it also used to mean that it just happens to be commutative for $x$ and $y$?
For example, is it correct usage to say, "1 and 1 commute under subtraction"?
What about, "$x$ and $y$ commute under subtraction when $x=y$"?
Edit: Additionally, is it incorrect usage to say a non-commutative operation is commutative under a set of circumstances, e.g. "subtraction is commutative when both operands are equal"?
 A: It is not incorrect to refer to a non-commutative operation as being commutative when operating on certain elements. In fact, this notion is very important! For instance, the center of a group is the set of elements that commute with every element of the group.
So the answer is a resounding "no". The algebraic structure need not be generally commutative to say "$x$ and $y$ commute." We must be careful, however, to make it clear that just because $x$ and $y$ commute, it does not mean that every element in our structure commutes.
A: All your examples are perfectly fine.
Another example is the set of $(n\times n)$-matrices equipped with multiplication. This multiplication is not commutative, but we say two matrices commute iff $AB=BA$. Or we could say "Matrix multiplication is commutative on diagonal matrices".
A: I think that everything you've said is fine.  It is trivially true that for any operation, $x$ and $y$ commute whenever $x=y$, so we wouldn't normally bother to say that.  But the property of commuting is a local property.  For example, in a group, the center of the group is the set of elements that commute with every element of the group. Since we always have $ex=xe$, it is trivial that the identity is a member of the center, and we would say that "for any $x$, the identity $e$ commutes with $x$").  
