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

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