# Definition:Minimal/Ordered Set

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$ be a subset of $S$.

### Definition 1

An element $x \in T$ is a **minimal element of $T$** if and only if:

- $\forall y \in T: y \preceq x \implies x = y$

That is, the only element of $T$ that $x$ **succeeds or is equal to** is itself.

### Definition 2

An element $x \in T$ is a **minimal element of $T$** if and only if:

- $\neg \exists y \in T: y \prec x$

where $y \prec x$ denotes that $y \preceq x \land y \ne x$.

That is, if and only if $x$ has **no strict predecessor**.

## Comparison with Smallest Element

Compare the definition of minimal element with that of a smallest element.

An element $x \in T$ is **the** smallest element of $T$ if and only if:

- $\forall y \in T: x \preceq y$

That is, $x$ is comparable to, and precedes, or is equal to, every $y \in T$.

An element $x \in T$ is **a** minimal element of $T$ if and only if:

- $y \preceq x \implies x = y$

That is, $x$ precedes, or is equal to, every $y \in T$ which *is* comparable to $x$.

If *all* elements are comparable to $x$, then such a minimal element is indeed **the smallest element**.

Note that when an ordered set is in fact a totally ordered set, the terms **minimal element** and **smallest element** are equivalent.

## Also defined as

Some treatments of this subject do not restrict the relation $\preceq$ to the requirement that it be an ordering.

## Also known as

Some sources refer to these as **atoms**.