Definition:Maximal/Element

Definition
Let $\left({S, \preceq}\right)$ be a poset.

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

An element $x \in T$ is a maximal element of $T$ iff:


 * $x \preceq y \implies x = y$

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

Alternatively, this can be put as:

$x \in T$ is a maximal element of $T$ iff:
 * $\neg \exists y \in T: x \prec y$

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

Comparison with Greatest Element
Compare this definition with that for a greatest element.

An element $x \in S$ is the greatest iff:


 * $\forall y \in S: y \preceq x$

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

Note that when a poset 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

 * Definition:Minimal Element