\begin{definition}[Definition:Auxiliary Relation]
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''' {{iff}}
{{begin-axiom}}
{{axiom | n = 1
        | q = \forall x, y \in S
        | m = \tuple {x, y} \in \RR \implies x \preceq y
}}
{{axiom | n = 2
        | q = \forall x, y, z, u \in S
        | m = x \preceq y \land \tuple {y, z} \in \RR \land z \preceq u \implies \tuple {x, u} \in \RR
}}
{{axiom | n = 3
        | q = \forall x, y, z \in S
        | m = \tuple {x, z} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x \vee y, z} \in \RR
}}
{{axiom | n = 4
        | q = \forall x \in S
        | m = \tuple {\bot, x} \in \RR
}}
{{end-axiom}}
where $\bot$ denotes the bottom and $\land$ denotes conjunction.
\end{definition}