Matrix representation of Boolean algebra?

Question

Is there such a thing as matrix representations of Boolean algebra?

Give a boolean algebra with finite elements {a,b,c...} and operations $\cap, \neg$, we can regard $\cap$ as matrix multiplication and we're free to define the "complement" of the matrices.

We want the matrix multiplication to preserve the structure of Boolean algebra. I learned that it is impossible for both matrix addition and multiplication to correspond to the boolean $\cap$ and $\cup$.

Is it possible to apply representation theory to find such matrices?

By the way, 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 $-x$.

Here I try to convert the axioms to multiplicative form and eliminate the additive operations:

1. $(ab)c = a(bc)$ --- associativity
2. $ab = ba$ --- commutativity
3. $aa = a$ --- idempotency
4. $-(-a) = a$
5. $a(-a) = 0$
6. $-1 = 0$
7. $0a = 0$

Axiom 3 implies that the matrices are projection matrices.

What else needs to be done?