Oh, so he did! I only noticed the first one, for some reason! Sorry, Dan!

The answer to my puzzle was going to be: "he chose \$$f \circ f = 1\_\star\$$ because he _felt like it_."

But anyway, \$$f \circ f = f\$$ also works fine. It gives a monoid with two elements that's not a group.

**Puzzle 102\$${}^{\prime\prime}\$$.** What's are some names for this monoid?