Well Inside Implies Predecessor

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Theorem

Let $L = \struct{S, \vee, \wedge, \preceq}$ be a distributive lattice with greatest element $\top$ and smallest element $\bot$.

Let $\eqslantless$ denote the well inside relation on $L$.

Then:

$\forall a,b \in S : a \eqslantless b \implies a \preceq b$

Proof

Let $a, b \in S : a \eqslantless b$.

By definition of well inside relation:

$\exists c \in S : a \wedge c = \bot, b \vee c = \top$

We have:

\(\ds a\)

\(=\)

\(\ds a \wedge \top\)

Predecessor is Infimum

\(\ds \)

\(=\)

\(\ds a \wedge \paren{b \vee c}\)

By choice of $c$

\(\ds \)

\(=\)

\(\ds \paren{a \wedge b} \vee \paren{a \wedge c}\)

Definition of Distributive Lattice

\(\ds \)

\(=\)

\(\ds \paren{a \wedge b} \vee \bot\)

By choice of $c$

\(\ds \)

\(=\)

\(\ds a \wedge b\)

Successor is Supremum

From Predecessor is Infimum:

$a \preceq b$

The result follows.

$\blacksquare$

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$