Order, lattices, ordered algebraic structures Definition

Infimum

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$ . An element $a$ of $P$ is an infimum, or greatest lower bound, of $S$ when these two conditions hold:

$a$ is a lower bound of $S$ ;
if $m$ is a lower bound of $S$ , then $m \le a$ .

The notation is $a = \inf S$ .

Remarks

A set has at most one infimum. In $\mathbb{R}$ , $\inf \{1/n : n \ge 1\} = 0$ , and $0$ is not an element of this set. For a subset $S$ of $\mathbb{R}$ , $\inf S = -\sup(-S)$ when one side exists, where $-S = \{-s : s \in S\}$ .

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Set bounded above definition
Infimum definition

Tag	`04KFTA34C5`
URI	`m2://04KFTA34C5`
Kind	definition
Writing	W0 stub
MSC	06A06
Defines	Infimum
Uses	Set bounded above

Infimum

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