\begin{definition}[Definition:Operator Generated by Ordered Subset]
Let $L = \struct {X, \precsim}$ be an ordered set.
Let $S = \struct {T, \preceq}$ be an ordered subset of $L$.
The '''operator generated by ordered set''' $S$ is defined by
:$\forall x \in X: \map {\map {\operatorname {operator} } S} x := \map {\inf_L} {x^\succeq \cap T}$
where $x^\succeq$ denotes the upper closure of $x$.
\end{definition}