Reduced product explained

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 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

\prod _{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 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 ^{n} _{i)/{\sim})} \iff \{i\in I\mid R ^{S _{i}} (a ^{n} _{i)\}\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

• Book: Chang. Chen Chung Chang . Chen Chung . Keisler . H. Jerome . Howard Jerome Keisler . Model Theory . 1973 . Elsevier . 3rd . Studies in Logic and the Foundations of Mathematics . 978-0-444-88054-3 . 1990 . ., Chapter 6.