Which associative binary operations on integers do you know, besides the $+,\cdot$? (It should have an identity element)
Motivation for asking this:
Define $E(n) = \sum_{d|n} d \log(d)$. Then for $gcd(m,n)=1$ we have the "Leibniz rule": $E(m \cdot n) = \sigma(m) \cdot E(n) + E(m) \cdot \sigma(n)$. So since the function composition $f \circ g$ is associative with identity element( a monoid ), I was wondering if additionaly to the Leibniz rule, if there is a chain rule for integers $(f \circ g)' = (f' \circ g) \cdot g'$. For this to make sense, I need a suitable monoid structure on the natural numbers $\circ$.
One infinite family would be $f(a,b)=a^nb^n$ for every natural $n$
Probably there is an infinite number of infinite families.