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Keisler in \cite{keislerordering} proved that if $\kappa$ is a
strong limit cardinal and $\lambda$ is a singular cardinal, then
the transfer relation $\kappa\longrightarrow\lambda$ holds. We
analyze the $\lambda$-like models produced in the proof of
Keisler's transfer theorem when $\kappa$ is further assumed to be
regular. Our main result shows that with this extra assumption,
Keisler's proof can be modified to produce a $\lambda$-like model
$M$ with built-in Skolem functions that satisfies the following
two properties:
(1) $M$ is generated by a subset $C$ of order-type $\lambda$.
(2) $M$ can be written as the union of an elementary end extension
chain $\langle N_{i}:i<\delta\rangle$ such that for each
$i<\delta$, there is an initial segment $C_{i}$ of $C$ with
$C_{i}\subseteq N_{i}$, and $N_{i}\cap(C\setminus
C_{i})=\emptyset$.
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