In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let {Si | i ∈ I} be a nonempty family of structures of the same signature σ indexed by a set I, and let U be a proper filter on I. The domain of the reduced product is the quotient of the Cartesian product
∏ i ∈ I S i {\displaystyle \prod _{i\in I}S_{i}}
by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if
{ i ∈ I : a i = b i } ∈ U {\displaystyle \left\{i\in I:a_{i}=b_{i}\right\}\in U}
If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the direct 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
R ( ( a i 1 ) / ∼ , … , ( a i n ) / ∼ ) ⟺ { i ∈ I ∣ R S i ( a i 1 , … , a i n ) } ∈ U . {\displaystyle R((a_{i}^{1})/{\sim },\dots ,(a_{i}^{n})/{\sim })\iff \{i\in I\mid R^{S_{i}}(a_{i}^{1},\dots ,a_{i}^{n})\}\in U.}
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.
References Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]. Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3., Chapter 6.