Definition:Infimum of Set

Definition
Let $\left({S, \preceq}\right)$ be a poset.

Let $T \subseteq S$.

An element $c \in S$ is the infimum of $T$ in $S$ if:


 * $(1): \quad c$ is a lower bound of $T$ in $S$
 * $(2): \quad d \preceq c$ for all lower bounds $d$ of $T$ in $S$.

The infimum of $T$ is denoted $\inf \left({T}\right)$.

The infimum of $x_1, x_2, \ldots, x_n$ is denoted $\inf \left\{{x_1, x_2, \ldots, x_n}\right\}$.

If there exists an infimum of $T$ (in $S$), we say that $T$ admits an infimum (in $S$).

Also defined as
Some sources refer to the infimum as being the lower bound. Using this convention, any number smaller than this is not considered to be a lower bound.

Also known as
The infimum of $T$ is often called the greatest lower bound of $T$ and denoted $\operatorname{glb} \left({T}\right)$.

Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.

Also see

 * Supremum


 * Supremum and Infimum Unique