Well Inside Relation Extends to Predecessor and Successor

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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:

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


Proof

Let $a,b,c,d \in S : a \preceq b \eqslantless c \preceq d$


By definition of well inside relation:

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


We have:

\(\ds \bot\) \(\preceq\) \(\ds a \wedge x\) Definition of Smallest Element
\(\ds \) \(\preceq\) \(\ds b \wedge x\) Infimum Precedes Coarser Infimum
\(\ds \) \(=\) \(\ds \bot\) By choice of $x$

From Ordering Axiom $(1)$: Reflexivity:

$a \wedge x = \bot$


Similarly, we have:

\(\ds \top\) \(=\) \(\ds c \vee x\) By choice of $x$
\(\ds \) \(\preceq\) \(\ds d \vee x\) Finer Supremum Precedes Supremum
\(\ds \) \(\preceq\) \(\ds \top\) Definition of Greatest Element

From Ordering Axiom $(1)$: Reflexivity:

$d \vee x = \top$


By definition of well inside relation:

$a \eqslantless d$


The result follows.

$\blacksquare$


Sources

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