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 an auxiliary relation if and only if

 $(1)$ $:$ $\displaystyle \forall x, y \in S:$ $\displaystyle \left({x, y}\right) \in \mathcal R \implies x \preceq y$ $(2)$ $:$ $\displaystyle \forall x, y, z, u \in S:$ $\displaystyle x \preceq y \land \left({y, z}\right) \in \mathcal R \land z \preceq u \implies \left({x, u}\right) \in \mathcal R$ $(3)$ $:$ $\displaystyle \forall x, y, z \in S:$ $\displaystyle \left({x, z}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x \vee y, z}\right) \in \mathcal R$ $(4)$ $:$ $\displaystyle \forall x \in S:$ $\displaystyle \left({\bot, x}\right) \in \mathcal R$

where $\bot$ denotes the bottom and $\land$ denotes conjunction.