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 ∈ 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.

If U is an ultrafilter, the reduced product is an ultraproduct.

Relations are interpreted by 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.

This mathematical logic-related article is a stub.