arithmetic
to organize the most common arithmetic operators, we use the hyperoperations as a semantic guide:
| n | hyperoperation | left inverse | right inverse | two-sided inverse |
|---|---|---|---|---|
| 1 | \(\href{https://en.wikipedia.org/wiki/Addition}{\operatorname{add}}(a,b) = c\) | \(\operatorname{lsub}(c,a) = b\) | \(\operatorname{rsub}(c,b) = a\) | \(\href{https://en.wikipedia.org/wiki/Subtraction}{\operatorname{sub}}(c, a) = b\ ∧\ \href{https://en.wikipedia.org/wiki/Subtraction}{\operatorname{sub}}(c,b) = a\) |
| 2 | \(\href{https://en.wikipedia.org/wiki/Multiplication}{\operatorname{mul}}(a,b) = c\) | \(\href{https://en.wikipedia.org/wiki/Division_(mathematics)#Left_and_right_division}{\operatorname{ldiv}}(c, a) = b\) | \(\href{https://en.wikipedia.org/wiki/Division_(mathematics)#Left_and_right_division}{\operatorname{rdiv}}(c, b) = a\) | \(\href{https://en.wikipedia.org/wiki/Division_(mathematics)}{\operatorname{div}}(c, b) = a\ ∧\ \href{https://en.wikipedia.org/wiki/Division_(mathematics)}{\operatorname{div}}(c, a) = b\) |
| 3 | \(\href{https://en.wikipedia.org/wiki/Exponentiation}{\operatorname{pow}}(a, b) = c\) | \(\href{https://en.wikipedia.org/wiki/Nth_root}{\operatorname{root}}(c, a) = b\) | \(\href{https://en.wikipedia.org/wiki/Logarithm}{\log}(c, b) = a\) | (not named yet) |
| 4 | \(\href{https://en.wikipedia.org/wiki/Tetration}{\operatorname{spow}}(a, b) = c\) | \(\href{https://en.wikipedia.org/wiki/Tetration#Super-root}{\operatorname{sroot}}(c, a) = b\) | \(\href{https://en.wikipedia.org/wiki/Tetration#Super-logarithm}{\operatorname{slog}}(c, b) = a\) | (not named yet) |
| … | … | … | … | … |
note: we do not include \(\operatorname{spow}\) \(\operatorname{sroot}\) \(\operatorname{slog}\) and other n ≥ 4 because there is no canonical definition yet (especially for non-integers)