Minimal element

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$. An element $m \in S$ is a minimal element of $S$ when no $s \in S$ satisfies $s < m$. Equivalently, for each $s \in S$, $s \le m$ implies $s = m$.

Remarks

It is the maximal element for the reversed order $\ge$. In $\{2, 3, 4, 6, 9\}$ under divisibility, the minimal elements are $2$ and $3$. A well-founded relation is one in which each nonempty subset has a minimal element.

Explanations

L1 human

A minimal element has nothing below it. Among the integers greater than $1$ under divisibility, the minimal elements are the primes.