User:IssaRice/Partial order summary table

Let $(X,\le )$ 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 $y\le M$ 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 $y_0\ge y$ for every $y\in Y$ (i.e. an upper bound which happens to be in the set)	Yes
Least upper bound	an upper bound $M$ such that if $K$ is another upper bound for $Y$, then $M\le K$	No
Supremum	(same as least upper bound, though in some cases like on the real line, the least upper bound is thought of as being a real number and will not exist when a set is not bounded above, whereas the supremum always exists)	No