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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"?

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up vote 6 down vote accepted

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.

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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".

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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 $x$").

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