Maximal element and greatest element

Let $(P, \le)$ be a poset (Partial order and partially ordered set) and $m \in P$.

$m$ is a maximal element of $P$ if no element of $P$ is strictly above it: there is no $x \in P$ with $m < x$. Equivalently, for every $x \in P$, if $m \le x$ then $x = m$.

$m$ is a greatest element (or maximum) of $P$ if $x \le m$ for every $x \in P$.

Minimal and least elements are defined dually, reversing every inequality.

• Maximal is not greatest, and the difference is the single most common confusion about ordered sets. A maximal element has nothing strictly above it; a greatest element is above everything. In a total order the two coincide, which is why the distinction is invisible to intuition trained on $\mathbb{R}$, but a partial order may have many maximal elements and no greatest one. The refutation is FALSE: every maximal element is a greatest element, witnessed by Two maximal elements and no greatest element ↗.
• A greatest element is always maximal, and it is unique when it exists: if $m$ and $m'$ are both greatest then $m \le m'$ and $m' \le m$, so $m = m'$ by antisymmetry. Maximal elements need not be unique.
• Zorn's lemma concludes that a maximal element exists, never that a greatest one does (Zorn's lemma). In a particular poset the maximal element it produces may happen to be greatest, since a greatest element is maximal; what Zorn never supplies is a guarantee of greatestness. Every application of Zorn therefore has to be phrased so that maximality is enough, typically by arranging the poset so that a maximal object cannot be extended, which is a statement about nothing being strictly above it.
• Maximality says nothing about comparability: a maximal element may be incomparable to other elements, including to other maximal ones. In Two maximal elements and no greatest element ↗ the two maximal elements are incomparable to each other, and in an antichain every element is maximal and is incomparable to all the rest.