Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital.

Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's truth value. E.g. $[x=x] = True$ for all $x$.

Let $o\colon B^n \to B$ a $n$-ary truth-operation, $0≠n\in \mathbb{N}$.

Let $B$ be a Rings $\mathfrak p\subset B$ a prime Ideal. We say the triple $(o,\mathfrak p,B)$ is $A$-guessable if $B$ is an $A$-Algebra (i.e. there is a chosen ring homomorphism $A\to B$) and there exists a polynomial $f_o=f_o(T_1,\ldots,T_n)\in A[T_1,\ldots,T_n]$ such that for all $x_1,\ldots, x_n\in B$ we have $[f_o(x_1,\ldots,x_n)\in\mathfrak{p}]=o([x_1\in \mathfrak{p}],\ldots,[x_n\in\mathfrak{p}])$.

Clearly for any Ring $A$ a triple $(o,\mathfrak p,B)$ is $B$-guessable (with $id\colon B\to B$) if it is $A$-guessable if it is $\mathbb{Z}$-guessable (with the obvious map $\mathbb{Z}\to B$).

Call a triple $(o, \mathfrak p,B)$ guessable if it is $B$-guessable

Call $o$ (very) easy if every triple $(o, \mathfrak p,B)$ is ($\mathbb{Z}$-) guessable.

Examples:

if $o=pr_i\colon(v_1,\ldots,v_n)\mapsto v_i$ for some $i$, then trivially every $(o, \mathfrak p,B)$ is guessable by $\mathbb{Z}$ via $f_o=T_i$

if $o=\vee\colon (v_1,\ldots,v_n)=v_1\vee\ldots\vee v_n$, then every $(o', \mathfrak p,B)$ is guessable by $\mathbb{Z}$ via $f_{o'}=T_1\cdot T_2\cdots T_n$. (because $\mathfrak{p}$ is prime)

So both $o$ and $o'$ are very easy.

**Question 1:** Are there other easy or very easy $n$-ary truth-operations other than $\vee$ and $pr$? In particular I would like to know if $\wedge$ is (very) easy.

**Question 2:** Is there a difference between guessable and $\mathbb{Z}$-guessable, i.e. are there triples which are guessable but not $\mathbb{Z}$-guessable? For which rings $B$ can such a difference occur?