Definition:Auxiliary Relation

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Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $\RR \subseteq S \times S$ be a relation on $S$.

Then $\RR$ is an auxiliary relation if and only if

\((1)\)   $:$     \(\ds \forall x, y \in S:\) \(\ds \tuple {x, y} \in \RR \implies x \preceq y \)             
\((2)\)   $:$     \(\ds \forall x, y, z, u \in S:\) \(\ds x \preceq y \land \tuple {y, z} \in \RR \land z \preceq u \implies \tuple {x, u} \in \RR \)             
\((3)\)   $:$     \(\ds \forall x, y, z \in S:\) \(\ds \tuple {x, z} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x \vee y, z} \in \RR \)             
\((4)\)   $:$     \(\ds \forall x \in S:\) \(\ds \tuple {\bot, x} \in \RR \)             

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