Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets
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Theorem
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.
Then:
- $\mathscr S$ is meet-continuous
- for every directed subsets $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$
Proof
Sufficient Condition
Let $\mathscr S$ be meet-continuous.
By Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals:
- for every ideals $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$
Let $D_1, D_2$ directed subsets of $S$.
By definition of up-complete:
- $D_1$ and $D_2$ admit suprema
By Supremum of Lower Closure of Set:
- $D_1^\preceq$ and $D_2^\preceq$ admit suprema
where
- $D_1^\preceq$ denotes the lower closure of $D_1$.
Thus
\(\ds \paren {\sup D_1} \wedge \paren {\sup D_2}\) | \(=\) | \(\ds \paren {\sup D_1^\preceq} \wedge \paren {\sup D_2^\preceq}\) | Supremum of Lower Closure of Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup \set {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}\) | $D_1^\preceq$ is ideal in $\mathscr S$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup \set {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}^\preceq\) | Supremum of Lower Closure of Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup \set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}^\preceq\) | Lower Closure of Meet of Lower Closures | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup \set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}\) | Supremum of Lower Closure of Set |
$\Box$
Necessary Condition
Assume that
- for every directed subsets $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$
By exemplification:
- for every ideals $I_1, I_2$ of $S$: $\paren {\sup I_1} \wedge \paren {\sup I_2} = \sup \set {d_1 \wedge d_2: d_1 \in I_1, d_2 \in I_2}$
Thus by Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals:
- $\mathscr S$ is meet-continuous.
$\blacksquare$
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_2:51