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:

$y \vee z \in x^\ll$

By Join Succeeds Operands:

$y \preceq y \vee z$ and $z \preceq y \vee z$

Thus by definition

$x^\ll$ is directed.

$\blacksquare$


Sources