Definition:Infimum

Ordered Set
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 $S$.

Plural: Infima.

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 $S$).

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

Mapping
Let $f$ be a mapping defined on a poset $\left({S, \preceq}\right)$.

Let $f$ be bounded below on $S$.

It follows from the Continuum Property that the codomain of $f$ has an infimum on $S$.

Thus:
 * $\displaystyle \inf_{x \in S} f \left({x}\right) = \inf f \left({S}\right)$

Also see

 * Supremum

Variants of Definition
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.

Sources for Ordered Set

 * : $\S 14$
 * : $\S 7$
 * : Definition $1.1.2$
 * : $\S 2.6$

Sources for Mapping

 * : $\S 7.13$