" tommy quaternion "

[MphLee]

The basic ideas are

1) unital and commutative but nonassociative numbers.
2) power-associative numbers so we can use taylor theorems.
3) no nilpotent elements
4) every element has at least 1 square root.
5) the smallest ones
6) no subnumbers only real coefficients. and not iso to an extension of 2 type of numbers ( like complex coefficients or other extensions of smaller dimensions )

then there are 2 cases left

the units sum to 0.

the units are linear independant.

assuming solutions exist ofcourse.  I conjecture yes.

On the other hand I conjecture only a finite amount of them ... probably between 0 and 3.
And all solutions having dimension below 28.

The 8 dimensional number given here has nilpotent elements. So it violates one of the conditions.
They always have a square root though.

I will post a candidate soon.

I was not able to find this relatively simple idea in the books.

I see applications in physics and math as I believe they are the " next quaternion ".

regards

tommy1729

That's more reasonable, but still not clear to me. Let's see if I get it...
Your are trying to axiomatize something, an agebraic structure that has the desired properties...
Those properties you want to enforce on them are motivated by your need to define functions valued in that structure that are enough well behaved for your purposes...
So given the properties, you are trying to pin down how many models those axioms have... how many isomorphic algebras have those properties

To be more precise you are looking for all the \(\mathbb R\)-algebras, i.e. \(\mathbb R\)-vector spaces \(V\) equipped with a bilinear product map \(\cdot:V\otimes V\to V\) that satisfies your list.

This seems a hell of an enterprise... I feel that classically (in the last 50/80 years) problems of this kind are treated via model theoretic methods, if not by universal algebra. And nowadays i feel that the way to examine this should be category-theoretic. For example, recently it was proved that a class of agebras interesting to physicians actually has only three models \(\mathbb R, \mathbb C\) and \(\mathbb H\) and it was proved using category theory if I remember good.

My question now is... why those properties? What do you want to do on those algebras? Extending tetration? Or is it a more vast programme?