I used to think about trying to generalize arithmetic operations like negation and addition. I never came to three but I did come up with the following operator that I thought was very interesting:
\( x \box y = \ln(e^x + e^y) \) which is holo in \( x \) and \( y \) by doing a little calculus that I'm too lazy to do atm ^_^.
It has the cool property \( (x \box y)+z = (x+z) \box (y+z) \)
and \( x + y \box x = x+ (y \box 0) \)
Then we define a nice metric:
\( ||x|| = |e^{x}| < e^{|x|} \)
so that
\( ||x \box y|| \le ||x|| + ||y|| \)
Now we see we can start talking about calculus even because this operator is continuous.
I always liked the box derivative:
\( \frac{\box}{\box x} f(x) = \lim_{h \to -\infty} [f(x\box h) \box (f(x) + \pi i)] - h \)
\( \frac{\box}{\box x}[ f(x) \box g(x)] = [\frac{\box}{\box x} f(x)] \box [\frac{\box}{\box x} g(x)] \)
\( \frac{\box}{\box x} nx = \ln(n) + (n-1)x \)
and general box analysis ^_^
This may be a little off topic, this thread just reminded me of this.
\( x \box y = \ln(e^x + e^y) \) which is holo in \( x \) and \( y \) by doing a little calculus that I'm too lazy to do atm ^_^.
It has the cool property \( (x \box y)+z = (x+z) \box (y+z) \)
and \( x + y \box x = x+ (y \box 0) \)
Then we define a nice metric:
\( ||x|| = |e^{x}| < e^{|x|} \)
so that
\( ||x \box y|| \le ||x|| + ||y|| \)
Now we see we can start talking about calculus even because this operator is continuous.
I always liked the box derivative:
\( \frac{\box}{\box x} f(x) = \lim_{h \to -\infty} [f(x\box h) \box (f(x) + \pi i)] - h \)
\( \frac{\box}{\box x}[ f(x) \box g(x)] = [\frac{\box}{\box x} f(x)] \box [\frac{\box}{\box x} g(x)] \)
\( \frac{\box}{\box x} nx = \ln(n) + (n-1)x \)
and general box analysis ^_^
This may be a little off topic, this thread just reminded me of this.