model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

, a branch of

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, and in

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the reduced product is a construction that generalizes both direct product and ultraproduct. Let be a nonempty family of structures of the same

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

σ indexed by a set ''I'', and let ''U'' be a proper filter on ''I''. The domain of the reduced product is the

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

of the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

\prod_ S_i

by a certain

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

~: two elements (''a_i'') and (''b_i'') of the Cartesian product are equivalent if :

\left\\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 In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...

, 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)/,\dots,(a^n_i)/) \iff \\in U.

For example, if each structure is a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

, 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

* , Chapter 6. Model theory {{Mathlogic-stub