Auxiliary Relation is Congruent
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Theorem
Let $\struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $\RR$ be relation on $S$ satisfying conditions $(2)$ and $(3)$ of auxiliary relation.
Then:
- $\forall x, y, z, u \in S: \tuple {x, z} \in \RR \land \tuple {y, u} \in \RR \implies \tuple {x \vee y, z \vee u} \in \RR$
Proof
Let $x, y, z, u \in S$ such that:
- $\tuple {x, z} \in \RR \land \tuple {y, u} \in \RR$
By definition of reflexivity:
- $x \preceq x$ and $y \preceq y$
- $z \preceq z \vee u$ and $u \preceq z \vee u$
By condition $(2)$ of auxiliary relation:
- $\tuple {x, z \vee u} \in \RR$ and $\tuple {y, z \vee u} \in \RR$
Thus by condition $(3)$ of auxiliary relation:
- $\tuple {x \vee y, z \vee u} \in \RR$
$\blacksquare$
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_4:1