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UltrapowerAn ultrapower is an important special case of the ultraproduct construction. Suppose κ is an infinite cardinal number and F is a structure in a first-order theory; for example F could be a field. A filter (mathematics) U on κ is a subset of the power set P(κ) not containing the empty set and closed under finite intersections, and such that any superset of an element of U is also an element of U. The filter is an ultrafilter if for any member of P(κ) either it or its complement is in U, and is a free ultrafilter if it contains no finite sets (which entails it contains all cofinite sets.) An ultrapower of F is a quotient Fκ/U for some free ultrafilter on U, where we now may extend the constants, functions, and relations in F to the equivalence classes of Fκ/U. If c is a constant in F, then we have a corresponding constant in the ultrapower as the equivalence class of the element of Fκ which is the constant function with constant c. If δ < κ is an ordinal number, and f is an element of Fκ, we may call , the element of F which δ is mapped to, the component of f at δ. If we have a relation, we have a corresponding relation on the ultrapower which holds if the set of components on which it holds is in U. Functions defined on F are also defined on the ultrapower by applying F to each component.
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