Many years ago, I considered so-called "numbers" whose multiplication is an alternating sum. For instance, a number "n" such that:
$m\cdot{n} = \underbrace{2+3+2+3+2+3...}_{\text{m}}$
For instance, $3n$ would be $2+3+2$ or $7$. But of course, such numbers would be impossible to produce over the reals. They would also conflict with rational numbers such as $\frac{7}{3}$, and no one number would satisify the condition above.
However, we could simply declare that these numbers exist, which I will refer to as alternating numbers. Obviously, such an algebra would have to include ony the integers $\mathbb{Z}$ and the new numbers, to avoid conflict. In this manner it can be thought of as an alternate extension of the integers, alas the p-adic numbers. For simplicity, I will denote these with angle brackets. We already know that, by definition:
$\langle{a,b}\rangle\cdot{c} = \underbrace{a+b+a+b+a+b...}_{\text{c}}$
This means that $\langle{a,a}\rangle$ is functionally equal to $a$, since $\langle{a,a}\rangle\cdot{c} = \underbrace{a+a+a+a+a+a...}_{\text{c}} = ac$ . Then we would need to define the multiplication of $\langle{a,b}\rangle$ and $\langle{c,d}\rangle$, so that it still holds the original identity $\langle{a,b}\rangle\cdot{c} = \underbrace{a+b+a+b+a+b...}_{\text{c}}$. This ad-hoc solution seems to work:
$\langle{a,b}\rangle\cdot{\langle{c,d}\rangle}= \langle{e, e + (d - c)}\rangle\\ \text{where e} = \underbrace{a+b+a+b+a+b...}_{\text{c}} $
The bias $d-c$ is equal to 0 when $\langle{c,d}\rangle$ is the integer $\langle{c,c}\rangle = c$, making the expression equal to $\langle{e,e}\rangle = e = \underbrace{a+b+a+b+a+b...}_{\text{c}}$, thus preserving the original definition.
As an example, $\langle{2,3}\rangle\cdot{\langle{2,4}\rangle} = \langle{5,7}\rangle$. But this definition of multiplication isn't very mathematically useful, as it is not assosciative or commutative, but it is still consistent with the multiplication of regular integers. Addition can be defined similarly:
$\langle{a,b}\rangle+{\langle{c,d}\rangle}= \langle{e, e + (d - c)}\rangle\\ \text{where e} = \underbrace{a@b@a@b@a@b...}_{\text{c}} $
Here, @ denotes zeration, or the operator @ such that $\underbrace{a\mathop @a\mathop @\ldots\mathop @ a}_n = a+n$. After some experimentation, I found out that $e$ is always equal to $b+c$, so we can further simplify the definition to:
$\langle{a,b}\rangle+{\langle{c,d}\rangle}= \langle{b+c, b + (d - c)}\rangle $
Once again, this operator does not appear to be assosciative or commutative. My questions are:
How do we define the subtraction and division operators? I can't seem to find any valid definition for them by simply treating them as the inverses of addition and multiplication. For instance, $\frac{17}{3}$ could be both, say, $\langle{5,7}\rangle$ or $\langle{6,5}\rangle$, so one would need to choose a "principle solution" to make these well-defined.
Is the system I described a group, semigroup, field or ring?
Can we create better definitions for addition and multiplication that are assosciative and/or commutative?
