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 {Si | i &isin; 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


 * $$\prod_{i \in I} S_i $$

by a certain equivalence relation ~: two elements (ai) and (bi) 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


 * $$R((a^1_i)/{\sim},\dots,(a^n_i)/{\sim}) \iff \{i\in I\mid R^{S_i}(a^1_i,\dots,a^n_i)\}\in U. $$

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.