Least element

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset 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. 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.