by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if
If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the original Cartesian product. If U is an ultrafilter, the reduced product is an ultraproduct.
Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are interpreted by
For example, if each structure is a vector space, then the reduced product is a vector space with addition defined as (a + b)i = ai + bi and multiplication by a scalar c as (ca)i = c ai.
- Chang, Chen Chung; Keisler, H. Jerome (1990) . Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3, Chapter 6.
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