论文标题
连接的还原产品
Connected Reduced Products
论文作者
论文摘要
如果$ρ$是集合$ x $的二进制关系，则结构$ {\ mathbb x} = \ langle x，如果将包含$ρ$的最小值的关系连接到$ x $的完全关系，则连接了ρ\ rangle $。我们表明，对于一套$ i $，以下条件是等效的 （a）$ | i | $小于第一个可测量的红衣主教 （b）对于每个过滤器$φ\ subset p（i）$和每个家庭$ \ {{\ mathbb x} _i：i \ in i \} $的二进制结构，还原产品$ \prod_φ{\prod_φ{\ mathbb x} _i $是连接的，如果连接了$ k \ s $ k \ s $ k \ subset，in $ {\ mathbb x} _i $是连接的，对于k $中的每个$ i \，以及$ \ {i \ in i：{\ mathbb x} _i \ mbox {是直径} \ leq n \ leq n \} （c）超副果$ \ prod _ {\ mathcal u} {\ mathbb g}_Ω$是每个非主要超级超级滤波器$ {\ mathcal u} \ subset p（i）$ {$ {\ mathbb g} _的$ umem $ $ $ goum的$。 此外，（b）中的含义“ $ \ leftarrow $”包含在ZFC中。
If $ρ$ is a binary relation on a set $X$, the structure ${\mathbb X}=\langle X,ρ\rangle$ is connected iff the minimal equivalence relation containing $ρ$ is the full relation on $X$. We show that, for a set $I$ the following conditions are equivalent (a) $|I|$ is less than the first measurable cardinal, (b) For each filter $Φ\subset P(I)$ and each family $\{ {\mathbb X}_i :i\in I\}$ of binary structures, the reduced product $\prod_Φ{\mathbb X}_i$ is connected, iff there are a finite set $K\subset I$ and $n\in ω$ such that ${\mathbb X}_i$ is connected, for each $i\in K$, and $\{ i\in I: {\mathbb X}_i \mbox{ is of diameter }\leq n\}\cup K\in Φ$, (c)The ultraproduct $\prod_{\mathcal U}{\mathbb G}_ω$ is a disconnected graph for each non-principal ultrafilter ${\mathcal U}\subset P(I)$, where ${\mathbb G}_ω$ is the linear graph on $ω$. Moreover, the implication "$\Leftarrow$" in (b) holds in ZFC.
