---
tag: 0Z7HDDXX3X
kind: definition
name: Greatest element
msc: [06A06]
defines: 0XR2QK49V6
uses: [011C29Q1S1]
sources:
  - url: https://en.wikipedia.org/wiki/Greatest_element_and_least_element
    locator: lead section
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] 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|0TZEEFTNCC]] and an [[upper bound|0H5J02P8S9]] 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 $6$.