Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
A D11: Set $E \subseteq X$ is downward enclosed in $P$ if and only if \begin{equation} \forall \, e \in E : \forall \, x \in X \left( (x, e) \in {\preceq} \quad \implies \quad x \in E \right) \end{equation}

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
A D11: Set $E \subseteq X$ is downward enclosed in $P$ if and only if \begin{equation} \forall \, e \in E : \forall \, x \in X \left( x \preceq e \quad \implies \quad x \in E \right) \end{equation}