Let $X$ be a D11: Set.
Let $\mathcal{P}(X)$ be the D80: Power set of $X$.
A D11: Set $\mathcal{S}$ is a subset algebra on $X$ if and only if $$\begin{equation} \mathcal{S} \subseteq \mathcal{P}(X) \end{equation}$$

Let $X$ be a D11: Set.
Let $\mathcal{P}^2(X)$ be a D4075: Hyperpower set of $X$.
A D11: Set $\mathcal{S}$ is a subset algebra on $X$ if and only if $$\begin{equation} \mathcal{S} \in \mathcal{P}^2(X) \end{equation}$$