\begin{definition}[Definition:Finite Suprema Set]
Let $P = \struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then '''finite suprema set''' of $X$, denoted $\map {\mathrm {finsups} } X$, is defined by:
:$\leftset {\sup A: A \in \map {\mathit {Fin} } X \land A}$ admits a supremum$\rightset{}$
where $\map {\mathit {Fin} } X$ denotes the set of all finite subsets of $X$.
\end{definition}