Definition:Approximating Relation

Definition

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

Let $\mathcal R$ be a relation on $S$.

Then $\mathcal R$ is approximating relation on $S$ if and only if

$\forall x \in S: x = \sup \left({x^{\mathcal R} }\right)$

where $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$.