  We develop further the consequences of the irreducible-Boolean classification
established in Ref. [9]; which have the advantage of allowing strong
statistical calculations in disordered Boolean function models, such as the
\textit{NK}-Kauffman networks. We construct a ring-isomorphism $ mathfrak{R}_K
{i_1, ..., i_\lambda} \cong \mathcal{P}^2 -[K] $ of the set of reducible
$K$-Boolean functions that are reducible in the Boolean arguments with indexes
${i_1, ..., i_\lambda}$; and the double power set $\mathcal{P}^2 [K]$, of the
first $K$ natural numbers. This allows us, among other things, to calculate the
number $\varrho_K (\lambda, \omega)$ of $K$-Boolean functions which are
$\lambda $-irreducible with weight $\omega$. $\varrho_K (\lambda, \omega)$ is a
fundamental quantity in the study of the stability of \textit{NK}-Kauffman
networks against changes in their connections between their Boolean functions;
as well as in the mean field study of their dynamics when Boolean
irreducibility is taken into account.
