handout02 - Math 5020 Handout 2 Ultraproducts and...

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Unformatted text preview: Math 5020 Handout 2 Ultraproducts and Ultrapowers Definition Let S be a set and F P ( S ). We say that F is a filter over S if (i) F ; (ii) if X F and X Y , then Y F ; (iii) if X, Y F then X Y F . A filter F over S is an ultrafilter if it is a maximal filter, i.e., if G is also a filter over S and F G , then G = F . Lemma A filter F over S is an ultrafilter iff for all X P ( S ) either X F or S X F . Lemma (AC) Any filter F can be extended an ultrafilter. Examples (1) The collection of all cofinite subsets of N is a filter over N . (2) If S is a nonempty set and a S , then the collection { A S : a A } is an ultrafilter over S (called the principal (ultra)filter generated by a ). Definition Let S be a set and X s be a set for each s S . The product of X s is s S X s = { x : S s S X s : x ( s ) X s s S } ....
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