Editing Tensor algebra (section)

=== Multiplication ===
Multiplication is given by an operator
:<math>\nabla: TV\boxtimes TV\to TV</math>
which, in this case, was already given as the "internal" tensor product. That is,
:<math>\nabla: x\boxtimes y\mapsto x \otimes y</math>
That is, <math>\nabla(x\boxtimes y) = x \otimes y.</math> The above should make it clear why the <math>\boxtimes</math> symbol needs to be used: the <math>\otimes</math> was actually one and the same thing as <math>\nabla</math>; and notational sloppiness here would lead to utter chaos.  To strengthen this: the tensor product <math>\otimes</math> of the tensor algebra corresponds to the multiplication <math>\nabla</math> used in the definition of an algebra, whereas the tensor product <math>\boxtimes</math> is the one required in the definition of comultiplication in a coalgebra.  These two tensor products are ''not'' the same thing!