关于布尔代数自由积的无限分拆的注记
A note on infinite partitions of free products of Boolean algebras
摘要：
如果$A$是一个无限布尔代数，则基数不变量$\mathfrak{a}(A)$被定义为$A$的无限划分的最小尺寸。基数$\mathfrak{a}(A\oplus B)$，其中$A\oplus B$是布尔代数$A$和$B$的自由积（其对偶拓扑空间是$A$和$B$的对偶拓扑空间的积），低于$\mathfrak{a}(A)$和$\mathfrak{a}(B)$。等式$\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$并不适用于所有无限布尔代数$A$和$B$。这里提供了$\mathfrak{a}(A\oplus B)$的一些下界。
英文摘要：
If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{a}(A\oplus B)$ are provided.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
