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

Revision as of 19:42, 2 October 2021

Let $(X, \leq)$ be a partially ordered set, and let $Y \subset X$ 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 $y_{0} \leq y$ for every $y \in Y$	Yes
Supremum		No
Least upper bound		No

@@ Line 1: / Line 1: @@
-Let <math>(X, \leq)</math> be a partially ordered set.
+Let <math>(X, \leq)</math> be a partially ordered set, and let <math>Y \subset X</math> be a subset of <math>X</math>.
 {| class="wikitable"
 |-
-! Term !! Definition !! Must be in set?
+! Term !! Definition !! Must be in set <math>Y</math>?
 |-
-| Upper bound || || No
+| Upper bound || <math>M \in X</math> such that <math>y \leq M</math> for all <math>y \in Y</math> || No
 |-
-| Maximal element || || Yes
+| 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 || || Yes
+| Maximum element || <math>y_0 \in Y</math> such that <math>y_0 \leq y</math> for every <math>y \in Y</math> || Yes
 |-
 | Supremum || || No