# Definition:Maximal/Ordered Set

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

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

### Definition 1

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

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

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

### Definition 2

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

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

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

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

## Comparison with Greatest Element

Compare the definition of maximal element with that of a greatest element.

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

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

That is, $x$ is comparable to, and succeeds, or is equal to, every $y \in S$.

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

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

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

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

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

## Also known as

Some sources refer to these as **coatoms** or **anti-atoms**.