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}

▾ Set of symbols

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Binary endorelation

▾ Preordering relation

▾ Partial ordering relation

▾ Partially ordered set

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Binary endorelation

▾ Preordering relation

▾ Partial ordering relation

▾ Partially ordered set

Formulation 0

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}

The

Formulation 1

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}

The

Dual definition