Maximum and Minimum

Definition

Let $(A,\le )$ be a partially ordered set and let $S\subseteq A$ be a subset.

A maximum of $S$ is an element $m\in S$ such that $s\le m$ for all $s\in S$. Equivalently, a maximum is a supremum that belongs to the set itself.

A minimum of $S$ is an element $m\in S$ such that $m\le s$ for all $s\in S$. Equivalently, a minimum is an infimum that belongs to the set itself.

Relationship to Supremum and Infimum

The maximum and minimum are special cases of supremum and infimum:

• If $S$ has a maximum $m$, then $m=\sup S$ and $m\in S$
• If $S$ has a minimum $m$, then $m=\inf S$ and $m\in S$
• A set may have a supremum without having a maximum (e.g., $(0,1)$ in $\mathbb{R}$ has $\sup (0,1)=1$ but no maximum)
• A set may have an infimum without having a minimum (e.g., $(0,1)$ in $\mathbb{R}$ has $\inf (0,1)=0$ but no minimum)

Existence and Uniqueness

When a maximum or minimum exists, it is unique by anti-symmetry of the partial order. However, not every subset has a maximum or minimum.

Maximal vs Maximum

A maximal element of $S$ is an element $x\in S$ such that there is no $y\in S$ with $x<y$. In a partial order, a set may have multiple maximal elements but at most one maximum.

• Every maximum is maximal
• A maximal element need not be a maximum (unless the order is total)
• If a set has a unique maximal element, it is the maximum

Dually, a minimal element of $S$ is an element $x\in S$ such that there is no $y\in S$ with $y<x$.

Notation

• Maximum: $\max S$ or $\max \{a_1,a_2,\dots ,a_n\}$
• Minimum: $\min S$ or $\min \{a_1,a_2,\dots ,a_n\}$
• For a pair: $\max (a,b)$ and $\min (a,b)$

Examples

• In $(\mathbb{R},\le )$, the set $[0,1]$ has maximum $1$ and minimum $0$
• In $(\mathbb{R},\le )$, the set $(0,1)$ has neither a maximum nor a minimum
• In $(\mathbb{N},\le )$, the set $\{2,4,6,8\}$ has maximum $8$ and minimum $2$
• In any totally ordered finite set, the maximum and minimum always exist
• In the power set $(\mathcal{P}(X),\subseteq )$, the set of all subsets has maximum $X$ and minimum $\varnothing$

Properties

• In a total order, every finite non-empty subset has both a maximum and a minimum
• A well-ordering is a total order in which every non-empty subset has a minimum
• For finite sets in any partial order, maximal elements exist, though they may not be unique

Related Concepts

• Supremum and Infimum
• Partial Order
• Total Ordering
• Well-Ordering
• Lattice
• Bound Lattice