let L be a bounded distributive lattice with dual space ( X := I p...
let L be a bounded distributive lattice with dual space , then the clopen downsets of X are .
Answer & Explanation
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 such that . Because D is compact, there is a finite subintersection which is a subset of D. A finite intersection of sets of the form is a set of the form , so for each I in D there is some a such that . Step 2 The union of all sets is therefore equal to D. Compactness yields a finite subunion, and a finite union of sets of the form is again a set of the form .
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