nLab maximal element

Idea

An element of a poset (or proset) is maximal if no other (inequivalent) element is greater. A maximum must be maximal, and a maximal element of a toset must be a maximum. However, it’s easy to find posets with maximal elements that aren't maxima, or even with a unique maximal element that isn't a maximum. The existence of a maximal element is often given by Zorn's lemma.

Definition

Let $P$ be a preordered set and $x$ an element of $P$. Then $x$ is maximal in $P$ if, whenever $x \leq y$ in $P$, we have $y \leq x$. Dually, $x$ is minimal in $P$ if, whenever $y \leq x$ in $P$, we have $x \leq y$.

Properties

Elementary properties

If $P$ has a top element, then this is the unique (up to equivalence) maximal element of $P$.

Suppose that $P$ is totally ordered. Then a maximal element of $P$ is the same as a top element of $P$.

Suppose that $P$ is finite and has a unique maximal element $x$. Then $x$ is a top element of $P$.

Deep properties

According to Zorn's Lemma, if every totally ordered subset of $P$ has an upper bound in $P$, then $P$ has a maximal element.

Examples

Let $P$ be $\{a,b,c\}$ with $a \leq b$, $a \leq c$, and no other nontrivial ordering. Then $b$ and $c$ are both maximal in $P$ (but of course not tops).

Let $P$ be the disjoint union of $\mathbb{N}$ (the poset of natural numbers) and a singleton {a}. Then $a$ is the unique maximal element of $P$ but still not a top.