In model theory, the reduced product construction contains a collection of structures or models, a set I that indexes the collection, and a filter U on I. Ultraproducts are a special case of reduced products, where U is an ultrafilter.
Boolean-valued models are used primarily in connection with independence proofs in set theory, for example in Rosser's book Simplified Independence Proofs. A boolean-valued model contains a boolean algebra B, and each sentence S of the language L is assigned an element of B as a value. These differ from standard models in which either S or ~S is satisfied at each model. Or one may say the standard models use a 2-valued boolean algebra.
Proposed construction: Take the index set I of a reduced product construction as a set of elements and give it the powerset boolean algebra P(I). Each sentence S of the language L is assigned an element of P(I), namely the set containing the indexes of the models at which it's satisfied. Use the filter U to define the quotient algebra P(I)/U. Then each sentence S of the language has a derived value in the quotient algebra. If the value assigned to S is in the filter U then S's value in the quotient algebra will be the unit element, and the value of ~S will be zero element. If U is an ultrafilter, every S will be asssigned the unit or the zero, but otherwise some sentences will have other values. This generates boolean-valued models for the language L.
This is only an outline; many details are missing. But it appears to show how to use a reduced product to construct a boolean-valued model. Does anyone know if this will or won't work, or has been done already?