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. $0 + a = 0$
 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 $-x$.