Algebra on Worlds

Given a structure $\mathfrak{A}$ there are many tuples of sequences of other structures $({\mathfrak{A}}_i{)}_{i\in I}$ and ultrafilters $\mathcal{U}$ s.t. $\mathfrak{A}={\prod }_{\mathcal{U}}{\mathfrak{A}}_i$. On the other hand, fixed the sequence, many ultrafilters can give rise to different ultraproducts. Fixed the sequence, an algebraic structure, i.e. a set of operations closed on the underlying set, on ultrafilters would define an algebraic structure on ultraproducts. Also, the algebraic structure on ultrafilter can be defined by an algebraic structure on generators of ultrafilters, i.e. sets which, once closed under intersection and supersets (this closure may entail a choice function that, for every subset $A$ of $\omega$, picks either $A$ or $\omega \setminus A$ so that the closure cannot be a non-maximal filter), become ultrafilters. My aim here is to find operations closed on the set of generators of ultrafilters so that those operations can be defined on the structures too.

The results I expect, once an algebraic structure on generators of ultrafilters is found, is to create a realm of research using the operations (depending on a chosen sequence of structures) of possible worlds, which may bring in the future to some developments in the semantics of modality and be the foundations for different modal logics. Perhaps further developments will enable a similar algebraic structure on possible worlds independent from the first choice of the sequence.

I took this claim as an assumption in Structure on Models and derived conclusions in the debate Semantic vs Syntactic View.