Directed in Join Semilattice with Finite Suprema
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Theorem
Let $\struct {S, \preceq}$ be a join semilattice.
Let $H$ be a non-empty lower section of $S$.
Then $H$ is directed if and only if
Proof
Sufficient Condition
Let us assume that
- $H$ is directed.
Let $A$ be a non-empty finite subset of $H$.
By Directed iff Finite Subsets have Upper Bounds:
- $\exists h \in H: \forall a \in A: a \preceq h$
By definition
- $z$ is upper bound of $A$
By Existence of Non-Empty Finite Suprema in Join Semilattice:
- $\sup A$ exists in $\struct {S, \preceq}$
By definition of supremum:
- $\sup A \preceq h$
Thus by definition of lower section:
- $\sup A \in H$
$\Box$
Necessary Condition
Let us assume that
Let $x, y \in H$.
By assumption:
- $\sup \set {x, y} \in H$
By definition of supremum:
- $\sup A$ is upper bound for $\set {x, y}$
Thus
- $x \preceq \sup A \land y \preceq \sup A$
Thus by definition
- $H$ 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_0:42