User:IssaRice/Partial order summary table: Difference between revisions

Revision as of 19:44, 2 October 2021

Let $(X, \leq)$ be a partially ordered set, and let $Y \subseteq X$ be a subset of $X$ .

Term	Definition	Must be in set $Y$ ?
Upper bound	$M \in X$ such that $y \leq M$ for all $y \in Y$	No
Maximal element	$y_{0} \in Y$ such that there exists no $y \in Y$ for which $y > y_{0}$	Yes
Maximum element	$y_{0} \in Y$ such that $y_{0} \geq y$ for every $y \in Y$ (i.e. an upper bound which happens to be in the set)	Yes
Supremum		No
Least upper bound		No

@@ Line 9: / Line 9: @@
 | Maximal element || <math>y_0 \in Y</math> such that there exists no <math>y \in Y</math> for which <math>y > y_0</math> || Yes
 |-
-| Maximum element || <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> || Yes
+| Maximum element || <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> (i.e. an upper bound which happens to be in the set) || Yes
 |-
 | Supremum || || No