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}

▾ 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.

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}

A D11: Set $E \subseteq X$ is

Formulation 1

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}

A D11: Set $E \subseteq X$ is

Dual definition

Also known as

Order ideal, Downset, Decreasing set