Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
The lower enclosure of $E \subseteq X$ in $P$ is the D11: Set \begin{equation} {\downarrow} E : = \bigcup_{e \in E} \{ x \in X : x \preceq e \} \end{equation}

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
The lower enclosure of $E \subseteq X$ in $P$ is the D11: Set \begin{equation} {\downarrow} E : = \{ x \in X : (\exists \, e \in E) x \preceq e \} \end{equation}