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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