---
tag: 0T2NTJCV83
kind: definition
name: Least element
msc: [06A06]
defines: 0AANKKNHS4
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 $l \in S$ is the least element of $S$ when $l \le s$ for each $s \in S$.

## Remarks

Another name is minimum. By antisymmetry, a set has at most one least element.
Each nonempty subset of $\mathbb{N}$ has a least element. This is the well-ordering principle of $\mathbb{N}$. See [[well-order|0F1D6PK9RZ]].
The interval $(0, 1]$ in $\mathbb{R}$ has no least element.

## Explanations

### L1 human

The least element is below all others.
In the power set of a set under $\subseteq$, the least element is the empty set.