A couple years ago I was playing with 2 different ideas at the same time: the fact that real addition is isomorphic to real positive multiplication, and the fact that the complex unit circle contains an isomorphic subgroup of each modular integer group.
This led directly to the development of the operation φ(a, b)=exp(ln(a)ln(b)), which when paired with multiplication forms a field over the positive reals that’s isomorphic to the reals. A slightly modified version: φ_β(a,b)=exp(ln(a)ln(b)×-i τ/β) (where β is a positive real (although I suspect anything other than τ or an integer is unlikely to be useful) and τ=2π) defines an infinite family of operations that forms a field over the circle group. I don’t think this qualifies as a real algebra, (and thus doesn’t contradict Zorn’s theorem) but it’s definitely a division algebra.
A couple years ago I was playing with 2 different ideas at the same time: the fact that real addition is isomorphic to real positive multiplication, and the fact that the complex unit circle contains an isomorphic subgroup of each modular integer group.
This led directly to the development of the operation φ(a, b)=exp(ln(a)ln(b)), which when paired with multiplication forms a field over the positive reals that’s isomorphic to the reals. A slightly modified version: φ_β(a,b)=exp(ln(a)ln(b)×-i τ/β) (where β is a positive real (although I suspect anything other than τ or an integer is unlikely to be useful) and τ=2π) defines an infinite family of operations that forms a field over the circle group. I don’t think this qualifies as a real algebra, (and thus doesn’t contradict Zorn’s theorem) but it’s definitely a division algebra.