Lower bound - Imported Knowlpedia Content

A lower bound of a subset $A$ of a partially ordered set $(P,\le)$ is an element $\ell\in P$ such that $\ell\le a$ for all $a\in A$ . The subset $A$ is bounded below if it has at least one lower bound in $P$ .

Remarks

Lower bounds complement upper bounds and lead to greatest lower bounds (the infimum in real analysis).

Examples

In $(\mathbb{Z},\le)$ , the integer $-5$ is a lower bound for the subset $\{-2,0,3\}$ .
In the poset $(\mathcal P(X),\subseteq)$ , the set $A\cap B$ is a lower bound for $\{A,B\}$ , where $A\cap B$ is the intersection of $A$ and $B$ .