Definition:Greatest/Ordered Set

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Definition

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 Element of Subset

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

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


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

$\forall y \in T: y \preceq \restriction_T x$

where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.


Comparison with Maximal Element

Compare the definition of 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 set 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.


Sources