Modulo operation in monotone complexity
In circuit complexity, a monotone operation is a function $f$ which is also monotone: if $x \leq y$ then $f(x) \leq f(y)$, where $x \leq y$ is a shorthand for $x_i \leq y_i$ for all $i$. There are many ways to define modulo $N$, but most of them are probably either uninteresting or not monotone. For example, consider the function which is true if the input is divisible by $N$. Then $f(0) = 1$ while $f(1) = 0$, and $0 \leq 1$ no matter what the actual length of the input is.