\begin{definition}[Definition:Maximal Element/Comparison with Greatest 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 $T$ {{iff}}:
:$\forall y \in T: y \preceq x$
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 $T$ {{iff}}:
:$x \preceq y \implies x = y$
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.
\end{definition}