In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect to addition ($-$ , $\neg$), inverse with respect to multiplication ($^{-1}$ , $\neg$) and constant symbols $0$, $1$ but these algebraic structures obey different axioms and the rules of inverse and identity are not completely match because in Boolean algebras $\neg$ doesn't show the properties of an inverse operator. e.g.
$\neg$ is "not" an inverse for $\vee$ because $\forall x\in\mathbb{B}~~~x\vee \neg x =1\neq 0$.
$\neg$ is "not" an inverse for $\wedge$ because $\forall x\in\mathbb{B}~~~x\wedge \neg x =0\neq 1$.
But there are similarities between $\neg$ and $^{-1}$, $-$ because $\neg$ sends each "variable" object $x$ of a Boolean algebra to a "constant" object but it does this in an "inverse" way with respect to the fields. For example $x\vee\neg x$ "should" be $0$ (because there is a natural correspondence between $\vee$ and $+$) but is $1$ and $x\wedge\neg x$ "should" be $1$ (because there is a natural correspondence between $\wedge$ and $\times$) but is $0$. Perhaps we need to revise our intuition about the "naturality" of the correspondences $\vee\leftrightarrow +$ and $\wedge\leftrightarrow \times$.
Question: Is there a natural way to assign a field $\mathbb{F}=\langle F,+,\times,-,^{-1},0,1\rangle$ to a given Boolean algebra $\mathbb{B}=\langle B,\vee,\wedge,\neg,0,1\rangle$? (Perhaps we should show a Boolean algebra with two different negation symbols one for a $\vee$-inverse operator and another for $\wedge$ as follows $\mathbb{B}=\langle B,\vee,\wedge,\neg_{\vee}, \neg_{\wedge},0,1\rangle$)
What about assigning a natural Boolean algebra to a field? By a "natural" way I mean something like category morphisms and quotient constructions, etc. Please introduce references for partial results.
Motivation: Forcing could be interpreted as a Boolean valued ultraproduct of the universe of all sets $V$. I am asking about the possibility of defining forcing using the structure of field. Another motivation comes from a similarity between the notion of "extending universe using a generic object" and "extending a field using a transcendental object".
The most natural algebraic object to associate with a Boolean algebra is a Boolean ring, in which the addition is the disjoint union $a\oplus b:= a\wedge \neg b\vee b\wedge\neg a$ and the multiplication is still $\vee$. Since $a\oplus a=0$ every Boolean ring is of characteristic $2$; the only Boolean ring that's a field is the field with two elements, since the only way $a\wedge b$ can be $1$ is if either $a$ or $b$ is.
It's possible to extend the notion of boolean-valued models to models valued in Heyting algebras, and further still; this hinges on the topological aspects of Boolean algebras more than the algebraic. Via related topos theoretic methods one can get Cohen's results in just a few pages, essentially by forcing but without using quite his terminology. On the other hand I think Asaf's objection indicates that you shouldn't expect a good notion of forcing with values in a field.