Way Below Relation is Auxiliary Relation
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Theorem
Lrt $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.
Then
- $\ll$ is auxiliary relation
where $\ll$ denotes the way below relation.
Proof
By Way Below implies Preceding:
- $\forall x, y \in S: x \ll y \implies x \preceq y$
By Preceding and Way Below implies Way Below:
- $\forall x, y, z, u \in S: x \preceq y \ll z \preceq u \implies x \ll u$
By Join is Way Below if Operands are Way Below:
- $\forall x, y, z \in S: x \ll z \land y \ll z \implies x \vee y \ll z$
By Bottom is Way Below Any Element:
- $\forall x: \bot \ll x$
where $\bot$ denotes the smallest element in $L$.
Thus by definition:
- $\ll$ is auxiliary relation.
$\blacksquare$
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:funcreg 20