Let \(B_n\) denote the multiplicative semigroup of matrices of order \(n\) over the Boolean algebra \(\{0,1\}\) and \(A = [a_{ij}] \in B_n\). The author studies the case of irreducible \(A\) and obtains the exact upper bound of weak exponents of any irreducible matrix \(A \in B_n\), namely, the weak primitive exponent \(e_w (A) \leq 2\); the weak fully indecomposable exponent \(f_w (A) \leq [{n \over 2}] + 1\); the weak Hall exponent \(h_w (A) \leq [{n \over 2}]\).
Furthermore, the corresponding exponent sets are determined, namely, the weak primitive exponent set is \(\{1,2\}\); the exponent set of \(f_w(A)\) is \(\{1,2, \ldots, [{n \over 2}] + 1\}\); the exponent set of \(h_w(A)\) is \(\{1,2, \ldots, [{n \over 2}]\}\).