  We use computer algebra to study the 512-dimensional associative algebra Q
B_3, the rational monoid algebra of 3 x 3 Boolean matrices. We obtain a basis
for the radical in bijection with the 42 non-regular elements of B_3. The
center of the 470-dimensional semisimple quotient has dimension 14; we use a
splitting algorithm to find a basis of orthogonal primitive idempotents. We
show that the semisimple quotient is the direct sum of simple two-sided ideals
isomorphic to d x d rational matrix algebras for d = 1, 1, 1, 2, 3, 3, 3, 3, 6,
6, 7, 9, 9, 12. We construct the irreducible representations of B_3 over Q by
calculating the representation matrices for a minimal set of generators.
