### Two Cryptomorphic Puzzles

#### Posted by John Baez

Here are two puzzles. One is from Alan Weinstein. I was able to solve it because I knew the answer to the other. These puzzles are ‘cryptomorphic’, in the vague sense of being ‘secretly the same’.

**Puzzle 1.** By ‘vector space’ let’s mean a finite-dimensional real vector space. Given orientations on vector spaces $A$ and $B$ there is a standard choice of orientation on their direct sum $A \oplus B$, which however depends on the fact that $A$ comes first and $B$ comes second. This choice makes the usual isomorphism

$(A \oplus B) \oplus C \cong A \oplus (B \oplus C)$

orientation-preserving. Can we go further, and also choose an orientation on the tensor product of any two oriented vector spaces in such a way that the usual isomorphisms

$A \otimes (B \oplus C) \cong (A \otimes B) \oplus (A \otimes C)$

$(A \oplus B) \otimes C \cong (A \otimes C) \oplus (B \otimes C)$

are orientation-preserving?

**Puzzle 2.** What would rings be like if we didn’t require that addition commute?

More precisely: what can you say about sets that have an group structure under addition and a monoid structure under multiplication, in such a way that multiplication distributes over addition on the right and left?

## Re: Two Cryptomorphic Puzzles

In response to the second puzzle: such structures would have commutative addition regardless, as we will have that (1 + a)(1 + b) equals both 1 + a + b + ab and 1 + b + a + ab, depending on the order in which one applies left and right distributivity; cancelling the 1 and ab terms by the group structure of addition, we are left with a + b = b + a.