Greatest Element/Examples
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Examples of Greatest Elements
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.