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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 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''. ==References== * , Chapter 6. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「reduced product」の詳細全文を読む スポンサード リンク
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