Idempotent that isn't any of these

Question

Let $A=\left\{x,y,z\right\}$ and $M=\left\{g\mid g:A\rightarrow A\text{ is a function}\right\}$.

Is there an element in $M$ that is idempotent but not right absorbing, left absorbing, a right identity, and a left identity? I've tried creating different functions (mainly piece-wise functions) that satisfy this, but I have been having trouble.

Edit: Would it be the empty function, which maps the empty set to A?

Answer 1

For $a \in A$, let $c_a$ be the constant function $c_a(x) = c_a(y) = c_a(z) = a$. Let $i: A \to A$ be the identity map, and let $f: A \to A$ be given by $f(x) = f(y) = x, f(z) = z$. Then:

1. $f$ is idempotent.
2. $f$ is neither left-absorbing nor a left identity, as $f \circ c_y = c_x$ (not $f$ or $c_y$).
3. $f$ is not a right identity, as $i \circ f = f$ (not $i$).
4. $f$ is not right-absorbing, as $c_a \circ f = c_a$ (not $f$) for any $a \in A$.