Determinants Associated to Zeta Matrices of Posets

We consider the matrix ${\mathfrak{Z}}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\mathfrak{Z}}_P$ and establish a recursive formula for this determinant in the case in which $P$ is a boolean algebra.