Characterization of Completely Prime Ideal in Complete Lattice

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Theorem

Let $\struct{L, \vee, \wedge, \preceq}$ be a complete lattice.

Let $I \subseteq L$.

Then:

$I$ is a completely prime ideal

if and only if

$(1)\quad\forall A \subseteq L : \bigwedge A \in I \iff \paren{\exists a \in A : a \in I}$

$(2)\quad\forall $ finite $A \subseteq L : \bigvee A \in I \iff \paren{\forall a \in A : a \in I}$

where:

• $\bigwedge A$ denotes the infimum of $A$ in $L$
• $\bigvee A$ denotes the supremum of $A$ in $L$

Proof

This is the dual statement of Characterization of Completely Prime Filter in Complete Lattice by Dual Pairs (Order Theory).

The result follows from the Duality Principle.

$\blacksquare$