Yes I saw the little mistake I made where I assumed \( f(0) = 1 \). The reason is because I've only really been observing functions which meet that requirement.
I think I'm not doing anything inconsistent but instead we have to create the law, which is not dissimilar to division by zero:
\( f(0)\,\otimes_f\,(a + b) = a + b \neq\, a\, \oplus_f\,b \)
or that the distributive law fails when f-multiplied by the identity.
I looked at that other thread too, very interesting. I had of hunch bo's proof but it's nice to see it proved.
but furthermore, this gives some very strange laws for multiplication:
\( v \,\otimes_f\,(k\cdot a) = v\,\otimes_f\,k\,\otimes_f\,a = k\,\otimes_f\,(v \cdot a) \)
which means for exponentiation:
\( v \,\otimes_f\,(a^k) = v \, \otimes_f\,(a^{k-1})\,\otimes_f\,a \)
which means f-multiplying a number to the power of another number we convert exponentiation to f-exponentiation:
\( v \,\otimes_f\,(a^k) = v\,\otimes_f\,(a^{\otimes_f\,k}) \)
which again is very very inconsistent. I must be doing something incorrect. I think we cannot give the distribution law, but that's not enough for me. I'd really like to know why. I'm absolutely puzzled.
I think I'm not doing anything inconsistent but instead we have to create the law, which is not dissimilar to division by zero:
\( f(0)\,\otimes_f\,(a + b) = a + b \neq\, a\, \oplus_f\,b \)
or that the distributive law fails when f-multiplied by the identity.
I looked at that other thread too, very interesting. I had of hunch bo's proof but it's nice to see it proved.
but furthermore, this gives some very strange laws for multiplication:
\( v \,\otimes_f\,(k\cdot a) = v\,\otimes_f\,k\,\otimes_f\,a = k\,\otimes_f\,(v \cdot a) \)
which means for exponentiation:
\( v \,\otimes_f\,(a^k) = v \, \otimes_f\,(a^{k-1})\,\otimes_f\,a \)
which means f-multiplying a number to the power of another number we convert exponentiation to f-exponentiation:
\( v \,\otimes_f\,(a^k) = v\,\otimes_f\,(a^{\otimes_f\,k}) \)
which again is very very inconsistent. I must be doing something incorrect. I think we cannot give the distribution law, but that's not enough for me. I'd really like to know why. I'm absolutely puzzled.