标题： Z_2-代数在布尔函数不可约分解中的应用 显示英文标题

标题： Z_2-Algebras in the Boolean Function Irreducible Decomposition

摘要： 我们进一步发展了在参考文献[9]中建立的不可约布尔分类的后果；这些后果的优点在于允许在无序布尔函数模型中进行强大的统计计算，例如\textit{NK}-Kauffman 网络。 我们构造了一个环同构$ mathfrak{R}_K {i_1, ..., i_\lambda} \cong \mathcal{P}^2 -[K] $，它是可约$K$-布尔函数的集合，这些函数在索引为${i_1, ..., i_\lambda}$的布尔变量中是可约的；以及前$K$个自然数的双重幂集$\mathcal{P}^2 [K]$。 这使我们能够计算$\varrho_K (\lambda, \omega)$个$K$-布尔函数的数量，这些函数在权重$\omega$下是$\lambda $-不可约的。 $\varrho_K (\lambda, \omega)$是研究\textit{NK}-Kauffman 网络在其布尔函数之间的连接发生变化时的稳定性的基本量；同时，在考虑布尔不可约性的情况下，也是研究其动力学的平均场理论中的基本量。 显示英文摘要

摘要： 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.