# 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.

## 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