Definition:Auxiliary Relation
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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