# Definition:Greatest

## Definition

### Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set.

An element $x \in S$ is the greatest element (of $S$) if and only if:

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

That is, every element of $S$ precedes, or is equal to, $x$.

The Greatest Element is Unique, so calling it the greatest element is justified.

Thus for an element $x$ to be the greatest element, all $y \in S$ must be comparable to $x$.

### Greatest Set

Let $\mathcal A$ be a set of sets or a class of sets.

Then a set $M$ is the greatest element of $\mathcal A$ (with respect to the subset relation) if and only if:

$M \in \mathcal A$ and
$\forall S: \paren {S \in \mathcal A \implies S \subseteq M}$