Ideal (order Theory) - Applications

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 sets are isomorphic to the original Boolean algebra.

• Order theory knows many completion procedures, to turn posets into posets with additional 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 dcpo generated by P. Furthermore the ideal completion serves to reconstruct any algebraic dcpo from its set of compact elements.