Definition:Finite Suprema Set

Definition
Let $P = \left({S, \preceq}\right)$ be an ordered set.

Let $X$ be a subset of $S$.

Then finite suprma set of $X$, denoted $\operatorname{finsups}\left({X}\right)$, is defined by
 * $\left\{ {\sup A: A \in \mathit{Fin}\left({X}\right) \land A}\right.$ admits a supremum$\left.{}\right\}$

where $\mathit{Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.