How to find matrix representations of a boolean algebra? [closed]

Question

Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.

How to find matrix representations of the elements such that:

1. boolean $\cup$ corresponds to matrix addition and
2. boolean $\cap$ corresponds to matrix multiplication?

Is it possible? If yes, is there a systematic way to find such matrices?

PS: the axioms of boolean algebra are:

1. $(a + b) + c = a + (b + c)$
2. $a + b = b + a$
3. $a + a = a$
4. $-(-b) = b$
5. $b + (-b) = 1$
6. $-1 = 0$
7. $0 + a = a$
8. $a \cdot (b+c) = a \cdot b + a \cdot c$
9. $a \cdot b \equiv -(-a + -b)$

where $\cup$ is denoted as + , $\cap$ as $\cdot$, and $\overline{x}$ as $-x$.

Answer 1

Let $B$ be a Boolean algebra and try to define the concept of a $B$-module in appropriate way (an abelian group of exponent 2 with suitable action of $B$ on it). Then you can see that such a $B$-module will be singleton. By this reason, the answer to your question is No.