This is my first question and I hope this question is not too brief to be acceptable:
There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably infinite sets. What if we combine the two properties?
Is there a bijective function $f\colon \mathbb{N \times N \to N}$ which is also associative: $f(x,f(y,z)) = f(f(x,y),z)\quad\text{ for all } \quad x,y,z \in \mathbb{N}\:?$
Thanks in advance!