Properties of Well Inside Relation

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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 the following hold:

Element Well Inside Itself Iff Has Complement

$\forall a \in S : a \eqslantless a \iff a$ has a complement

Well Inside Implies Predecessor

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

Well Inside Relation Extends to Predecessor and Successor

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

Well Inside Elements Form Filter

$\forall a \in S : \set{b \in S: a \eqslantless b}$ is a lattice filter

Elements Well Inside Form Ideal

$\forall a \in S : \set{b \in S: b \eqslantless a}$ is a a lattice ideal

Sources

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