Order n-transitivity

An ordered permutation group is order n-transitive if it can send any increasing $n$-tuple to any other increasing $n$-tuple. This compares tuples with the same relative order, unlike unrestricted n-transitive group actions. In the usual transitive chain setting, order two-transitivity implies every finite order-transitivity for a lattice-ordered permutation group.

Suppose $h_{ij}$ sends the $i$th and $j$th nodes of an increasing input tuple to the corresponding output nodes. At the $i$th input, every $h_{ij}$ agrees, so their pointwise minimum takes the required value. At every other input it is bounded above by the required value, using the corresponding $j$ term. The pointwise maximum over $i$ therefore interpolates the entire tuple. The construction uses closure under finite join and meet, without assuming arbitrary restrictions of group elements belong to the group.