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 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
by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if
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
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.
Famous quotes containing the words reduced and/or product:
“The Gettysburg speech is at once the shortest and the most famous oration in American history. Put beside it, all the whoopings of the Websters, Sumners and Everetts seem gaudy and silly. It is eloquence brought to a pellucid and almost gem-like perfection—the highest emotion reduced to a few poetical phrases.”
—H.L. (Henry Lewis)
“The history is always the same the product is always different and the history interests more than the product. More, that is, more. Yes. But if the product was not different the history which is the same would not be more interesting.”
—Gertrude Stein (1874–1946)