Lower bound

Let $(P,\le)$ be a partially ordered set and let $S\subseteq P$ be a subset . An element $l\in P$ is a lower bound of $S$ if

Lower bounds are dual to upper bounds. In ordered settings, one often asks whether a greatest lower bound exists (the infimum ).

Examples:

• In $(\mathbb{R},\le)$, the number $0$ is a lower bound of $(0,1)$ but is not an element of $(0,1)$.
• In $(\mathcal{P}(X),\subseteq)$, the intersection of a family of subsets is a lower bound of that family.
• In $(\mathbb{Z},\le)$, the set $\{n\in\mathbb{Z}: n\ge 0\}$ has no lower bound.