An ultrafilter is a special kind of filter on a set that contains all the supersets of its elements and is maximal in a certain sense, meaning it cannot be extended by adding more sets without losing its filter properties. This concept is crucial when working with ultraproducts and ultrapowers as it provides a way to focus on certain subsets of a structure while maintaining a coherent sense of convergence. Ultrafilters help streamline complex structures by allowing the formation of equivalence classes that preserve certain properties, which plays a significant role in understanding model behavior.
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