# alpha stable

Suppose $\alpha: X \times X \to X$. A subset $A \subseteq X$ is called $\alpha$-stable if and only if $x,y \in A \Rightarrow \alpha(x,y) \in A$.
What is the point of considering $\psi_{\alpha}(\psi_{\alpha}(A)) = \psi_{\alpha}(A)$? Is there any significance?
What is $\psi_\alpha$ ?