While reading a probability paper titled Gröbner bases and factorisation in discrete probability and Bayes (can't find free version, sorry), I came across the explanation:
The set of indicator functions $I_i$ form a Boolean algebra that is in one-to-one correspondence with the quotient of the ring $\mathbb{R}[x_1, \ldots, x_n]$ with respect to the relationships $x_i(x_i - 1)=0$ where $x_i$ represents $I_i$.
$I_i$ is defined to be the indicator function $I_i(\omega) = 1 \;\textrm{if}\; \omega \in A_i \;\textrm{else}\; 0$, $i = 1, \ldots, n$.
I have a rudimentary understanding of rings, but can you please explain the correspondence shown here?
Cheers
Edit: In the rest of the paper, the authors don't seem to limit themselves to $\mathbb{F}_2$. In fact, I can't see a need for the use of the minus sign in $x_i(x_i-1)$ if the field is $\mathbb{F}_2$. I can't verify if the coefficients of the polynomials are disregarded yet.