Well (to Dan Asimov) First: you & Gerver, you said, were using real division algebras. You did not say how. If you did not need the full strength of the distributive law, e.g. only need right-distributivity, then you could use my 16-ons. My question to you, was: can you? Second: you said the definition of "division algebra" is well established and my answer is: that may be true in your mind... but if you actually look in the literature, I have found verbatim definitions under which my 16-ons are one. Even though they are proven not to be one. By a famous impossibility theorem. The reason is the literature definitions tend to be careless about the distributive law, they sort of assume it without explicitly saying so, and so forth. I suspect they had in mind what you do, but the fact is, their verbatim words say otherwise in at least some cases. Basically, it has occurred to very few people, ever, to think about (what I call) "nonlinear algebras." That is understandable since they are hard to deal with. But in the specific case of my "2^N-on" things, I was able to do it since they have a nice structure. There might be other nonlinear algebras with nice structure waiting to be found -- who knows. Certainly not me. But I was able to prove some stuff, like the only normed ones have to be power-of-2 dimensional.