# Greatest lower bound

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Given a subset $S$ in some larger partially ordered set $R$, a greatest lower bound or infimum for $S$ is an element $m \in R$ such that $m \leq s$ for every $s \in S$ and there is no $M > m$ with this same property.

If the greatest lower bound $m$ of $S$ is an element of $S$, it is also the minimum of $S$. If $m \not\in S$, then $S$ has no minimum.