Maximal element

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$. An element $m \in S$ is a maximal element of $S$ when no $s \in S$ satisfies $m < s$. Equivalently, for each $s \in S$, $m \le s$ implies $s = m$.

Remarks

A maximal element need not be the greatest element, and a set can have several maximal elements. In $\{\{1\}, \{2\}, \{1, 2, 3\}, \{4\}\}$ under $\subseteq$, the maximal elements are $\{1, 2, 3\}$ and $\{4\}$. Zorn's lemma gives maximal elements in many infinite posets.

Explanations

L1 human

A maximal element has nothing above it. It may still be incomparable with other elements, so it is not necessarily above everything.