Editing Ideal (order theory) (section)

==Applications==

The construction of ideals and filters is an important tool in many applications of order theory.
* In [[Stone's representation theorem for Boolean algebras]], the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a [[topological space]], whose [[clopen set]]s are [[isomorphism|isomorphic]] to the original Boolean algebra.
* Order theory knows many [[completion (order theory)|completion procedures]] to turn posets into posets with additional [[completeness (order theory)|completeness]] properties. For example, the [[ideal completion]] of a given partial order ''P'' is the set of all ideals of ''P'' ordered by subset inclusion. This construction yields the [[free object|free]] [[directed complete partial order|dcpo]] generated by ''P''. An ideal is principal if and only if it is [[Compact element|compact]] in the ideal completion, so the original poset can be recovered as the sub-poset consisting of compact elements. Furthermore, every [[algebraic poset|algebraic dcpo]] can be reconstructed as the ideal completion of its set of compact elements.