Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in literature a same simple formalization of interrogative affirmations, that is questions, as a universal algebra. There is a pretty large domain of research, erotetic logic, but I did not encounter a simple "algebra of questions", same simple as Boolean algebra. This sounds strange to me due to the argument below.
In one of his publications in computer science, Donald Knuth treated questions as ideals of a Boolean algebra - a treatment which sounds very natural. Namely, he treated a question $Q$ as the set of all answers to it, and such answers make up an ideal. Really, for any two propositions $X$ and $Y$ the following two conditions are satisfied: (a) if $X$ implies $Y$, and $Y$ replies to question $Q$, then it is natural to consider $X$ also an anwer to $Q$. (b) if both $X$ and $Y$ reply to $Q$, then also $ X\lor Y$ reply to $Q$. This leads to the idea that questions make up a special universal algebra correlated with a boolean algebra and Stone theorem must play a role in defining this algebra and this correlation. Due its strong correlation with Boolean algebra, this univeral algebra must play same key role in algebra as the Boolean algebra, and not only serve the role of formalizing the questions.
Is there any research of an "algebra of questions" treated as ideals of a Boolean algebra?
What kind of universal algebra make up the ideals of a Boolean algebra?
- If such universal algebras are defined, is there a theorem "inverse" to Stone representation theorem, which allow for such an algebra $A$ to build a Boolean algebra $B$ "of replies to questions from $A$"?
I am aware that some of my questions sound naive.