# Definition:Auxiliary Relation

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## Definition

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.

## Sources

- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott:
*A Compendium of Continuous Lattices*

- Mizar article WAYBEL_4:def 7