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 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

[math]\displaystyle{ \prod_{i \in I} S_i }[/math]

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

[math]\displaystyle{ \left\{ i \in I: a_i = b_i \right\}\in U }[/math]

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

[math]\displaystyle{ R((a^1_i)/{\sim},\dots,(a^n_i)/{\sim}) \iff \{i\in I\mid R^{S_i}(a^1_i,\dots,a^n_i)\}\in U. }[/math]

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.