Definition:Auxiliary Relation

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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.