(i) | $X \neq \emptyset$ |

**minimum element**in $P$ if and only if \begin{equation} \forall \, x \in X : (m, x) \in {\preceq} \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

▾ Minimal element

▾ 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

▾ Minimal element

Formulation 0

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

A D2218: Set element $m \in X$ is a **minimum element** in $P$ if and only if
\begin{equation}
\forall \, x \in X : (m, x) \in {\preceq}
\end{equation}

(i) | $X \neq \emptyset$ |

Formulation 1

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

A D2218: Set element $m \in X$ is a **minimum element** in $P$ if and only if
\begin{equation}
\forall \, x \in X : m \preceq x
\end{equation}

(i) | $X \neq \emptyset$ |

Formulation 2

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

A D2218: Set element $m \in X$ is a **minimum element** in $P$ if and only if
\begin{equation}
m \preceq X
\end{equation}

(i) | $X \neq \emptyset$ |

Dual definition

Also known as

Least element, Bottom element, Orderwise zero

Child definitions

Results