Peter Selinger wrote yesterday:

I know three monoidal structures on the category of sets, all of them symmetric. Two are the product and coproduct, and I'll leave it to your imagination to figure out the third one.

My question is: are these the only three? Proofs, counterexamples, or references appreciated.

Several people took up the challenge, and provided me with interesting monoidal structures on Set. I'll briefly summarize the correct responses that I have received. Let me begin by revealing the "third" monoidal structure that I had in mind. It is X*Y = XY + X + Y. (1) It is easy to see that this satisfies the axioms if one rewrites it, by an abuse of notation, as X*Y = (X+1)(Y+1) - 1. Steve Lack pointed out that this is part of an infinite family of monoidal structures, each defined by X*Y = XSY + X + Y, (2) where S is some fixed set. Of course, for S=0 this is just the coproduct, and for S=1, this is the same as (1). In the case |S| > 1, coherence is not totally obvious; in fact, there are two possible natural isomorphisms (X*Y)*Z -> X*(Y*Z), deriving from the two natural maps from S^2 to itself. Only one of them is coherent. The coherence proof is somewhat simpler if one writes (2) in the form I have given, rather than in Steve's original form SXY + X + Y. It is interesting to note that, contrary to appearances, Steve's monoidal structure is not symmetric (not even braided) for |S| > 1. There is only one candidate braiding map X*Y -> Y*X, and it fails to satisfy the hexagon axiom. Ralph Loader proposed another monoidal structure, not contained in Steve's family: let X*Y be the set of non-empty finite sequences in X+Y, with no two consecutive elements from the same component of X+Y. Using the Kleene star, this can be written as X*Y = (XY)^* (X+XY) + (YX)^* (Y+YX). (3) (Here, A^* is the list monad, i.e., the initial solution for A^* = 1 + A A^*, also known as the Kleene star). After Ralph saw Steve's family, he noticed that his construction can also be generalized to an infinite family, by alternating list elements with elements of S, namely: X*Y = (XSYS)^* (X+XSY) + (YSXS)^* (Y+YSX). (4) The case S=0 is again the coproduct, and the case S=1 is of course (3). Unlike Steve's family, these tensors appear to be symmetric for all S. Also unlike Steve's family, the construction does not restrict to the category of finite sets. Jeff Egger contributed another symmetric monoidal structure, which he calls "par", and which is defined by: X*Y = X if Y=0 X*Y = Y if X=0 (5) X*Y = 1 if both X and Y are non-empty. This can be uniquely extended to morphisms such that f*0 = f = 0*f. It is perhaps interesting to note that an attempt to make Jeff's construction into an infinite family, by replacing "1" by some pointed set S, *almost* succeeds: the resulting operation is functorial, with coherent associativity and unit isomorphism. The only problem is that associativity fails to be a natural transformation. I mention it here because it might make for a really neat exercise in a course. In summary, we have two infinite families (2) and (4) (both including coproduct), plus product and Jeff's "par" (5). I admit that I did not expect so many non-trivial monoidal structures to exist on Set, and I now expect that there are many more. A complete classification would be interesting, but is perhaps too much to expect. I will close with another challenge: consider only *symmetric* monoidal structures on the category of *finite* sets. So far, we have seen four such structures, namely product, coproduct, and the tensors (1) and (5). Are these the only four? -- Peter P.S. monoidal *closed* structures, as mentioned in Steve's message, are an entirely different ball game. The requirement that tensor is left adjoint implies that it preserves colimits in each components; on Set, everything is therefore determined by 1*1. The only possibility is 1*1=1, which yields the usual cartesian-closed structure.