Supremum and Infimum

Definition

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

An upper bound for $S$ is an element $u\in A$ such that $s\le u$ for all $s\in S$. The supremum (or least upper bound) of $S$, denoted $\sup S$, is an upper bound $x\in A$ such that $x\le u$ for every upper bound $u$ of $S$.

Dually, a lower bound for $S$ is an element $l\in A$ such that $l\le s$ for all $s\in S$. The infimum (or greatest lower bound) of $S$, denoted $\inf S$, is a lower bound $x\in A$ such that $l\le x$ for every lower bound $l$ of $S$.

Formal Characterization

The supremum $x=\sup S$ satisfies:

1. $s\le x$ for all $s\in S$ (upper bound property)
2. For all $z\in A$, if $s\le z$ for all $s\in S$, then $x\le z$ (least property)

The infimum $y=\inf S$ satisfies:

1. $y\le s$ for all $s\in S$ (lower bound property)
2. For all $w\in A$, if $w\le s$ for all $s\in S$, then $w\le y$ (greatest property)

Existence and Uniqueness

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

A complete lattice is a partial order in which every subset has both a supremum and an infimum.

Notation

• Join: For a pair $\{a,b\}$, the supremum is written $a\vee b$
• Meet: For a pair $\{a,b\}$, the infimum is written $a\wedge b$
• General supremum: $\sup S$ or $\bigvee S$
• General infimum: $\inf S$ or $\bigwedge S$

Categorical Perspective

In the thin category corresponding to a poset, the supremum of a set $S$ is its coproduct, and the infimum of $S$ is its product.

Examples

• In $(\mathbb{R},\le )$, the set $(0,1)$ has $\sup (0,1)=1$ and $\inf (0,1)=0$, neither of which are in the set
• In $(\mathbb{N},\le )$, the set $\{2,4,6,8\}$ has $\sup \{2,4,6,8\}=8$ and $\inf \{2,4,6,8\}=2$
• In $(\mathcal{P}(X),\subseteq )$, the supremum is union $\bigcup$ and the infimum is intersection $\bigcap$
• In the divisibility order on $\mathbb{N}$, $\sup \{a,b\}$ is the least common multiple and $\inf \{a,b\}$ is the greatest common divisor

Related Concepts

• Lattice
• Complete Lattice
• Partial Order
• maximum
• Coproduct (Category Theory)
• Product (Category Theory)
• Thin Category