Way Below Closure is Directed in Bounded Below Join Semilattice
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Theorem
Let $\struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $x \in S$.
Then
- $x^\ll$ is directed.
Proof
By Bottom is Way Below Any Element:
- $\bot \ll x$
By definition of way below closure:
- $\bot \in x^\ll$
Thus by definition:
- $x^\ll$ is a non-empty set.
Let $y, z \in x^\ll$
By definition of way below closure:
- $y \ll x$ and $z \ll x$
By Join is Way Below if Operands are Way Below
- $y \vee z \ll x$
By definition of way below closure:
- $y \vee z \in x^\ll$
- $y \preceq y \vee z$ and $z \preceq y \vee z$
Thus by definition
- $x^\ll$ is directed.
$\blacksquare$
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_3:funcreg 1
- Mizar article WAYBEL_3:funcreg 4