nLab
upper bound

Definitions

Definition

For $(S,\le )$ a preorder and $A\hookrightarrow S$ a subset, an upper bound of $A$ in $S$ is an element $x$ of $S$ such that $y\le x$ whenever $y\in A$.

Dually (in the opposite poset?):

Definition

For $(S,\le )$ a preorder and $A\hookrightarrow S$ a subset, a lower bound of $A$ in $S$ is an element $x$ of $S$ such that $x\le y$ whenever $y\in A$.

Related concepts

• Zorn's lemma