> there's a reason that the rules of algebra are what they are
But which algebra, and which rules ? There are infinitely many algebras ( eg. Algebra over the field of reals, Banach Algebra, relational algebra, boolean algebra, sigma algebra etc. ) The "rules" are really constructs you decide that apply to the elements of the space that conform to your algebra. So for example the reals are a field that have ordering, so you can talk about less than and greater than, but the complex numbers don't have an imposed order and you'd have to first define a norm to map them onto the reals. The AltDragonBox with its own inconsistent arbitrary rules will still have some algebraic encoding. Whether that's useful to you is debatable. Like in my algebra I could overload plus to mean multiply and square root to mean divide by 7 and add -3 and then try to figure out what exponentiation works out to. It would be interesting...maybe not useful, but its still an algebra. Maybe you won't have closure...the elements may not end up in a field or even in a semigroup...its a nice make-believe algebra.
Yes, that's a very impressive display of all the math you must surely know, but its completely misses the point. The rules of elementary algebra really are special, and it has to do with their correspondance to real things in real life. There's a reason mathematicians don't just enumerate all possible algebras and study them one by one.
But which algebra, and which rules ? There are infinitely many algebras ( eg. Algebra over the field of reals, Banach Algebra, relational algebra, boolean algebra, sigma algebra etc. ) The "rules" are really constructs you decide that apply to the elements of the space that conform to your algebra. So for example the reals are a field that have ordering, so you can talk about less than and greater than, but the complex numbers don't have an imposed order and you'd have to first define a norm to map them onto the reals. The AltDragonBox with its own inconsistent arbitrary rules will still have some algebraic encoding. Whether that's useful to you is debatable. Like in my algebra I could overload plus to mean multiply and square root to mean divide by 7 and add -3 and then try to figure out what exponentiation works out to. It would be interesting...maybe not useful, but its still an algebra. Maybe you won't have closure...the elements may not end up in a field or even in a semigroup...its a nice make-believe algebra.