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A linear order $(L, <)$ is $\lambda$-transitive iff any order-preserving bijection between sets of size $\lambda$ can be extended to an order automorphism of $L$.

For $\lambda < \aleph_0$, $2$-transitive orders can be found as the reduct of an ordered field and $2$-transitive implies $k$-transitive for all $k < \aleph_0$. I'm curious about what happens when $\lambda$ is infinite. In particular, do $\lambda$-transitive linear orders (or something like them) exist?

Some additional restriction seems necessary: different order types or map between a sequence with a lub and one without would cause problems. If the definition requires the sets to be inf and sup closed, does this help?

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    $\begingroup$ Any first-order structure has a strongly $\lambda^+$-homogeneous elementary extension, and any strongly $\lambda^+$-homogeneous dense linear order is $\lambda$-transitive in your terminology. $\endgroup$ Apr 2, 2015 at 18:40
  • $\begingroup$ (Conversely, note that a 2-transitive order with at least 3 elements has to be dense.) $\endgroup$ Apr 2, 2015 at 19:45
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    $\begingroup$ What is an "increasing set"? $\endgroup$
    – bof
    Apr 2, 2015 at 21:25
  • $\begingroup$ That, too. In the comment above, I interpreted “increasing bijection between increasing sets” as “order-preserving bijection between sets”. $\endgroup$ Apr 2, 2015 at 22:13
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    $\begingroup$ That is, the following are equivalent for any structure $M$: (1) every partial self-embedding of size $\le\lambda$ extends to an automorphism; (2a) $M$ is strongly $\lambda^+$-homogeneous, and (2b) every partial self-embedding [of size $\le\lambda$, makes no difference] is elementary. If $M$ has quantifier elimination, (2b) holds automatically. For linear orders, (2b), quantifier elimination, and density are equivalent. If you insist that the $\lambda$-transitivity condition only applies to sets of size exactly $\lambda$, rather than at most $\lambda$, the only difference it will make ... $\endgroup$ Apr 7, 2015 at 18:48

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