Here’s one possible solution to the puzzle I posted last week. The task was to find a commutative and associative function with no identity element. I had hoped to post a more clever solution, but found some small errors in edge-cases and technicalities to everything I had planned on posting! But still, here’s a solution that does work, even if it’s a little ugly:
$$
f(x,y)=\left\{ \begin{array}{rl}
x+y &\mbox{ if }x \neq 0, y \neq 0 \\
0 &\mbox{ otherwise}
\end{array} \right.
$$
Or similarly:
$$
f(x,y)=\left\{ \begin{array}{rl}
xy &\mbox{ if }x \neq 1, y \neq 1 \\
1 &\mbox{ otherwise}
\end{array} \right.
$$