Suppose $\displaystyle \alpha: X \times X \to X $. A subset $\displaystyle A \subseteq X $ is called $\displaystyle \alpha $-stable if and only if $\displaystyle x,y \in A \Rightarrow \alpha(x,y) \in A $.

What is the point of considering $\displaystyle \psi_{\alpha}(\psi_{\alpha}(A)) = \psi_{\alpha}(A) $? Is there any significance?