TITLE: What other monoidal structures exist on the category of sets?
QUESTION [36 upvotes]: I know of the following monoidal structures over $\mathbf{Set}$ (taken from here):

The Cartesian product $\otimes=\times$ (categorical product)
The disjoint union $\otimes=+$ (categorial coproduct)
$A\otimes B = A\times S\times B + A + B$ for some given set $S$
$A\otimes B = (A\times S\times B\times S)^* \times (A + A\times S \times B) + (B\times S\times A\times S)^* \times (B + B\times S \times A)$ where $*$ is the Kleene star (free monoid construction)
If $A=\emptyset$ then $A\otimes B=B$, if $B=\emptyset$ then $A\otimes B=A$, otherwise $A\otimes B=I$

I want to know if there are any more that are known or, conversely, if there are any demonstrations that additional examples don't exist.
Also, what if we restrict to, say, braided or symmetric monoidal categories, or traced monoidal categories (or some example), are there any results/examples regarding the existence/non-existence of monoidal structures over $\mathbf{Set}$ in these cases?

REPLY [3 votes]: Here is another possible source of tensor products on $\mathbf{Set}$: transferring tensor products from categories $\mathbf{Set}$ is coreflective in. Unfortunately I have no good example.
Suppose $(\mathbf{C},\otimes,I)$ is a monoidal category with a coreflection from $\mathbf{Set}$; that is, there are functors $F\colon \mathbf{Set} \to \mathbf{C}$ and $G \colon \mathbf{C} \to \mathbf{Set}$ such that $F$ is left adjoint to $G$ with isomorphic unit $\eta_X \colon X \to GF(X)$. Moreover, suppose that $FG \colon \mathbf{C} \to \mathbf{C}$ is monoidal. Then $X \otimes Y := G(FX \otimes FY)$ is a monoidal product on $\mathbf{Set}$. (See e.g. Theorem 5 here.)
For example, these conditions are automatically fulfilled if $\mathbf{C}$ is the category of topological spaces and $G$ is the forgetful functor. However, a quick browse gives results classifying the tensor products on $\mathbf{Top}$ (*), so this is probably not a good source of examples. Are there others?
(*) That are canonical - meaning they just give the tensor product $\times$ on $\mathbf{Set}$ - and closed; but we are precisely looking for noncanonical ones...