Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.

Then:

$\mathscr S$ is meet-continuous

if and only if:

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

Retrieved from "https://proofwiki.org/w/index.php?title=Meet-Continuous_iff_Meet_of_Suprema_equals_Supremum_of_Meet_of_Directed_Subsets&oldid=664924"