Maximal Element/Examples
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Examples of Maximal Elements
Finite Subsets of Natural Numbers
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.
Consider the ordered set $\struct {\FF, \subseteq}$.
There are no maximal elements of $\struct {\FF, \subseteq}$.
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 $\struct {\GG, \subseteq}$.
There are no maximal elements of $\struct {\GG, \subseteq}$.