Definition:Operator Generated by Ordered Subset

Definition
Let $L = \left({X, \precsim}\right)$ be an ordered set.

Let $S = \left({T, \preceq}\right)$ be an ordered subset of $L$.

The operator generated by ordered set $S$ is defined by
 * $\forall x \in X:\operatorname{operator}\left({S}\right)\left({x}\right) := \inf_L\left({x^\succeq \cap T}\right)$

where $x^\succeq$ denotes the upper closure of $x$.