TITLE: Usage of "commute"/"commutative"
QUESTION [5 upvotes]: 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"?

REPLY [3 votes]: 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 $x$").