Minimal and Maximal

Topics: Discrete Mathematics - Partially Ordered Set

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(definition)

Let $(A,⪯)$ be a poset.

• An element $m\in A$ is a maximal element of $(A,⪯)$ if there isn’t any $x\in A$ such that $m\prec x$
• An element $s\in A$ is a minimal element of $(A,⪯)$ if there isn’t any $x\in A$ such that $x\prec s$

Examples

• $(\mathbb{R},\le )$ doesn’t have any minimal or maximal elements.
• $({\mathbb{R}}_{+},\le )$ has $0$ as its minimal element (if ${\mathbb{R}}_{+}=[0,\infty )$)
• $(\mathbb{N},\le )$ has $0$ as its minimal element
• $(\mathcal{P}(X),\subseteq )$ has $\varnothing$ as its minimal and $X$ as its maximal
• $$\begin{gathered} (\mathcal{P}(X), \\ \varnothing ) \end{gathered}$$ has $\{a\},\{b\},\{c\}$ as its minimals, and $X$ as its maximals

Unique Maximal

We have can only have a unique maximal element in totally ordered sets.

There can also be absolute minimal and maximal elements in posets.