It is well-known that a category in which finite products exist and coincide with finite coproducts (i.e. a zero object exists and the binary product functor is naturally equivalent to the binary coproduct functor) is semi- additive, i.e. enriched over the monoidal closed category of commutative monoids. Conversely, in a semi- additive category, every (existing) finite product is automatically a finite coroduct in the canonical way (an vice versa). Now my student Claus Kirschner observed that the proof of the latter fact does not use associativity and commutativity of the addition. Strange enough, together with the first fact this means: If a category with finite products has an operation called "addition" with neutral element 0 on each hom-set satisfying f0=0f=0 and the distributive laws, then the addition is automatically associative and commutative. Has anyone ever seen this before? Of course, finite products (or coproducts) are essential; otherwise one can easily construct single-object counterexamples. On the other hand, Hilbert observes something similar but different in an appendix to his book "Grundlagen der Geometrie": For unital rings, the commutativity of addition follows from the other axioms; one can even relax the exisence of the additive inverse to cancellation conditions. The argument is as follows: x+x+y+y=(1+1)x+(1+1)y=(1+1)(x+y)=1(x+y)+1(x+y)=x+y+x+y Cancelling x on the left and y on the right, we get x+y=y+x. Does anybody know more about similar implications? Greetings Reinhard