Definition:Maximal Element/Comparison with Greatest Element

Definition
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 $T$ :


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

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 $T$ :


 * $x \preceq y \implies x = y$

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.