Reduced product

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 {S_i | i ∈ I} be a family of structures of the same signature σ indexed by a set I, and let U be a filter on I. The domain of the reduced product is the quotient of the Cartesian product

$\prod _{{i\in I}}S_{i}$

by a certain equivalence relation ~: two elements (a_i) and (b_i) of the Cartesian product are equivalent if

$\left\{i\in I:a_{i}=b_{i}\right\}\in U\,$

If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the 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

$R((a_{i}^{1})/{\sim },\dots ,(a_{i}^{n})/{\sim })\iff \{i\in I\mid R^{{S_{i}}}(a_{i}^{1},\dots ,a_{i}^{n})\}\in U.\,$

For example, if each structure is a vector space, then the reduced product is a vector space with addition defined as (a + b)_i = a_i + b_i and multiplication by a scalar c as (ca)_i = c a_i.

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.