User:IssaRice/Partial order summary table

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