# Definition:Maximal/Element

## Definition

Let $\struct {S, \RR}$ be a relational structure.

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

### Definition 1

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

- $x \mathrel \RR y \implies x = y$

### Definition 2

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

- $\neg \exists y \in T: x \mathrel {\RR^\ne} y$

where $x \mathrel {\RR^\ne} y$ denotes that $x \mathrel \RR y$ but $x \ne y$.

### Class Theory

In the context of class theory, the definition follows the same lines:

Let $A$ be a class.

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

- $\forall y \in A: x \not \subset y$

That is, $x$ a proper subset of no element of $A$.

## Comparison with Greatest Element

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

Consider the ordered set $\struct {S, \preceq}$ such that $T \subseteq S$.

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 with, 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 with $x$.

If *all* elements are comparable wth $x$, 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 defined as

Most treatments of the concept of a **maximal element** restrict the definition of the relation $\RR$ to the requirement that it be an ordering.

However, this is not strictly required, and this more general definition as used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is of far more use.

## Also known as

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

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## Examples

### Finite Subsets of Natural Numbers

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Consider the ordered set $\struct {\FF, \subseteq}$.

There are no maximal elements of $\struct {\FF, \subseteq}$.

### Finite Subsets of Natural Numbers less Empty Set

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Let $\GG$ denote the set $\FF \setminus \O$, that is, $\FF$ with the empty set excluded.

Consider the ordered set $\struct {\GG, \subseteq}$.

There are no maximal elements of $\struct {\GG, \subseteq}$.

## Also see

- Results about
**maximal elements**can be found**here**.