Lydia Carey

Answered

2022-06-27

let L be a bounded distributive lattice with dual space $(X:={\mathcal{I}}_{p}(L),\subseteq ,\tau )$, then the clopen downsets of X are ${X}_{a},a\in L$.

Answer & Explanation

Jaylee Dodson

Expert

2022-06-28Added 22 answers

Step 1

Consider a prime ideal I in a clopen downset D. The downset generated by I is the intersection of all the sets ${X}_{a}$ such that $a\notin I$. Because D is compact, there is a finite subintersection which is a subset of D. A finite intersection of sets of the form ${X}_{a}$ is a set of the form ${X}_{a}$, so for each I in D there is some a such that $I\in {X}_{a}\subseteq D$.

Step 2

The union of all sets ${X}_{a}\subseteq D$ is therefore equal to D. Compactness yields a finite subunion, and a finite union of sets of the form ${X}_{a}$ is again a set of the form ${X}_{a}$.

Consider a prime ideal I in a clopen downset D. The downset generated by I is the intersection of all the sets ${X}_{a}$ such that $a\notin I$. Because D is compact, there is a finite subintersection which is a subset of D. A finite intersection of sets of the form ${X}_{a}$ is a set of the form ${X}_{a}$, so for each I in D there is some a such that $I\in {X}_{a}\subseteq D$.

Step 2

The union of all sets ${X}_{a}\subseteq D$ is therefore equal to D. Compactness yields a finite subunion, and a finite union of sets of the form ${X}_{a}$ is again a set of the form ${X}_{a}$.

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