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 $\sim S$ 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 $\sim S$ 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?