# Category:Definitions/Suprema

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This category contains definitions related to Suprema.
Related results can be found in Category:Suprema.

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$.

An element $c \in S$ is the supremum of $T$ in $S$ if and only if:

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

If there exists a supremum of $T$ (in $S$), we say that:

$T$ admits a supremum (in $S$) or
$T$ has a supremum (in $S$).