Join is Way Below if Operands are Way Below

Theorem

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

Let $x, y, z \in S$ such that

$x \ll z$ and $y \ll z$

where $\ll$ denotes the way below relation.

Then

$x \vee y \ll z$

Proof

Let $D$ be a directed subset of $S$ such that

$D$ admits a supremum

and

$z \preceq \sup D$

By definition of way below relation:

$\exists d_1 \in D: x \preceq d_1$

and

$\exists d_2 \in D: y \preceq d_2$

By definition of directed subset:

$\exists d \in D: d_1 \preceq d$ and $d_2 \preceq d$

By definition of transitivity:

$x \preceq d$ and $y \preceq d$

Thus by definition of supremum:

$x \vee y \preceq d$

Thus by definition way below relation:

$x \vee y \ll z$

$\blacksquare$

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_3:3