# Definition:Greatest Element/Class Theory

## Definition

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be an ordering.

Let $A$ be a subclass of the field of $\RR$.

An element $x \in A$ is the greatest element of $A$ if and only if:

$\forall y \in A: y \mathrel \RR x$

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

## Comparison with Maximal Element

Compare the definition of a 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 known as

The greatest element of a collection is also called:

• The largest element (or biggest element, etc.)
• The last element
• The maximum element (but beware confusing with maximal - see above)
• The unit element (in the context of boolean algebras and boolean rings)

## Examples

### Finite Subsets of Natural Numbers

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Consider the ordered set $\struct {\FF, \subseteq}$.

$\struct {\FF, \subseteq}$ has no greatest element.

### 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 $\struct {\GG, \subseteq}$.

$\struct {\FF, \subseteq}$ has no greatest element.

## Also see

• Results about greatest elements can be found here.