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{\bf Products of $\kappa$-Suslin Trees} \\
{Sy D. Friedman}\\
{Mathematics Department, MIT}\\
{sdf@math.mit.edu}
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We extend Jensen's work on $\kappa$-Suslin trees in $L$ to the study of
sequences of such trees. Assume $V=L$. Suppose that $\lambda$ is a limit
of limit cardinals and that for each limit cardinal $\lambda_0 < \lambda,
T_{\lambda_0}$ is a $\lambda_0^+$-Suslin tree. The \emph{reduced product} of 
the $T_{\lambda_0}, \lambda_0 < \lambda$ is the product of the $T_{\lambda_0},
\lambda_0 < \lambda$, factored by the equivalence relation which identifies
two elements of this product when they agree on a final segment of the
$\lambda_0$'s. 

{\bf Theorem} Assume $V=L$. There exist $\langle T_\lambda | \lambda$ a limit 
cardinal$\rangle$ such that $T_\lambda$ is a $\lambda^+$-Suslin Tree for each 
$\lambda$ and when $\lambda$ is a limit of limit cardinals, $T_\lambda$ is 
embeddable into the reduced product of the $T_{\lambda_0}, \lambda_0 < 
\lambda$. 

One can extract from the proof a combinatorial principle which relates the
$\diamondsuit$ and $\square$ sequences at different limit cardinals to each
other, via a nice scale.

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