Element Well Inside Itself Iff Has Complement

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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:

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

Proof

Follows immediately from:

• Definition:Well Inside Relation
• Definition:Complement (Lattice Theory)

$\blacksquare$

Sources

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