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

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

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

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

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

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

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