Definition:Auxiliary Relation

Definition
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

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

Then $\mathcal R$ is auxiliary relation
 * $(i): \quad \forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x \preceq y$ and
 * $(ii): \quad \forall x, y, z, u \in S: x \preceq y \land \left({y, z}\right) \in \mathcal R \land z \preceq u \implies \left({x, u}\right) \in \mathcal R$ and
 * $(iii): \quad \forall x, y, z \in S: \left({x, z}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x \vee y, z}\right) \in \mathcal R$ and
 * $(iv): \quad \forall x \in S: \left({\bot, x}\right) \in \mathcal R$