# Definition:Greatest

## Definition

### Ordered Set

Let $\struct {S, \preceq}$ 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 $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.

Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by $\subseteq$ considered as an ordering.

Then $T \in \TT$ is the greatest set of $\TT$ if and only if $T$ is the greatest element of $\struct {\TT, \subseteq}$.

That is:

$\forall X \in \TT: X \subseteq T$