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 $m^{\mathrm{\prime }}$ are both greatest then $m\le m^{\mathrm{\prime }}$ and $m^{\mathrm{\prime }}\le m$, so $m=m^{\mathrm{\prime }}$ 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.