Greatest element

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$. An element $g \in S$ is the greatest element of $S$ when $s \le g$ for each $s \in S$.

Remarks

Another name is maximum. By antisymmetry, a set has at most one greatest element. A greatest element is a maximal element and an upper bound of $S$ that lies in $S$. The interval $[0, 1)$ in $\mathbb{R}$ has no greatest element.

Explanations

L1 human

The greatest element beats every other element of the set. In $\{1, 2, 3, 6\}$ under divisibility, $6$ is the greatest element, because each element divides $6$.