Definition:Maximal/Element

From ProofWiki
Jump to navigation Jump to search

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 $x \ne y$.


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.



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.