Do you know if the following set of rules has an algebra more general than the usual complex numbers?
Maybe someone can also help me state the rules in some mathematically rigorous (fancy :) ) way so that I have a solid statement.
The elements are objects indexed with two real numbers. The multiplication is associative and satisfies
$X_{a,b}\cdot X_{c,d}=X_{ac,b+d}$
Moreover the second index is periodic
$X_{a,b+1}=X_{a,b}$
[EDIT: from further research it seems I actually define a 'field' and require the two above rules]
Now I always require that some operation called addition satisfying all axioms of common addition (associative, commutative, zero, negation) yields an element within the set of these elements
[EDIT: I think I forgot the rule $X_{a,0}+X_{b,0}=X_{a+b,0}$ or is it not needed?]
$\exists e,f:X_{a,b}+X_{c,d}=X_{e,f}$
The multiplication from above is distributive over this addition.
Do these rules already define how $e$ and $f$ should be? I have a shaky proof that the have to be complex number rules, but I'm not sure if there is something more general. One shaky part is that I require square roots.
(Extra: what if I remove the condition about periodicity in the second index?)
[EDIT: Motivation of this algebra: I imagined the first parameter being some sort of growth/scaling parameter and the second parameter as some sort of aging parameter. The aging parameter is periodic. The rules mentioned are supposed to logically represent growth and aging. Addition relates to growth. Now I wondered whether addition is uniquely determined if I assume that for some reason this a sum should always give back a single growth and a single aging.] ]