Order, lattices, ordered algebraic structures Definition

Supremum

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$ . An element $b$ of $P$ is a supremum, or least upper bound, of $S$ when these two conditions hold:

$b$ is an upper bound of $S$ ;
if $M$ is an upper bound of $S$ , then $b \le M$ .

The notation is $b = \sup S$ .

Remarks

A set has at most one supremum, see uniqueness of the supremum. In $\mathbb{R}$ , $\sup [0, 1) = 1$ and $\sup [0, 1] = 1$ . The first set does not contain its supremum. In $\mathbb{Q}$ , the set $\{q \in \mathbb{Q} : q^2 < 2\}$ is bounded above but has no supremum.

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Set bounded above definition
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Tag	`0BTMQ04556`
URI	`m2://0BTMQ04556`
Kind	definition
Writing	W0 stub
MSC	06A06
Defines	Supremum
Uses	Set bounded above

Supremum

Definition

Remarks

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