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# Dismal arithmetic

## Definition

Dismal arithmetic is a commutative algebra in which operations on single digits are defined by[1]

${\displaystyle a\oplus b=\max\{a,b\},a\otimes b=\min\{a,b\}\,}$

or equivalently

${\displaystyle a\oplus b=\lim _{k\to \infty }{\sqrt[{k}]{a^{k}+b^{k}}},a\otimes b=\lim _{k\to \infty }{\frac {1}{\sqrt[{k}]{{\tfrac {1}{a^{k}}}+{\tfrac {1}{b^{k}}}}}}\,}$

## Notes

1. [ALS], p. 2.