Definition:Maximal/Ordered Set

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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 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.



Also see