Non-Empty Way Below Closure is Directed in Join Semilattice

Theorem

Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.

Let $x \in S$ such that:

$x^\ll \ne \O$

where $x^\ll$ denotes the way below closure of $x$.

Then $x^\ll$ is directed.

Proof

By assumption:

$x^\ll$ 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

• 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 4