Lattice-ordered permutation group

A group of order automorphisms of a totally ordered set that is closed under pointwise maximum and minimum. Products act on the right when writing $\alpha f$; then $\alpha (fg)=(\alpha f)g$. The pointwise lattice operations make it a lattice-ordered group.

A transitive lattice-ordered permutation group is order-primitive when its only invariant equivalence relations with convex classes are equality and the universal relation. This concerns convex blocks in a chain, rather than arbitrary blocks of a primitive group action.

A faithful transitive order-primitive lattice permutation group is either a regular Archimedean translation action, an order two-transitive action, or a periodic action. In the periodic case a fixed-point-free order automorphism of the Dedekind completion commutes with the action and has cofinal and coinitial iterates; the point stabilizers act order two-transitively within period intervals. Only the regular case satisfies $|f||g|\le |g{|}^2|f{|}^2$ for every pair. In either other case, order two-transitivity on the whole chain or on a period interval supplies positive $a,b$ and $x<y<z<t$ with $xa=y$, $ya=z$, $xb=x$, $yb=t$, giving $xab=t>z=x{b}^2{a}^2$. The classification itself is a theorem, not a consequence of that witness calculation.

Every lattice-ordered group embeds in a lattice-ordered permutation group on one totally ordered set. Choose a value in a lattice-ordered group for each nonidentity element, act on its ordered right cosets, and combine these chains in an order sum. The original element moves its own value's identity coset, making the action faithful. The coset lattice formulas make the embedding preserve both join and meet.

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.

A positive bump of the real line is an order automorphism whose nonempty support of a permutation is one open interval and which moves every point of that interval to the right. It fixes the complement. Iterating any interior point tends to the right endpoint in positive time and to the left endpoint in negative time.

Choose fundamental intervals $[x,xf)$ and $[y,yg)$ for two positive bumps of an order automorphism. Any increasing bijection matching their endpoints extends by $b(x{f}^n)=(xb)g^n$ to their support intervals. Extend it monotonically over the two complementary rays. Then $fb=bg$, so $b^{-1}fb=g$. Bounded supports impose no further invariant for positive bumps in the full real-line order-automorphism group.

If two positive bumps of an order automorphism move a point $c$ beyond an interval $[c,d]$, their conjugacy can be chosen to fix that interval: begin with the identity there, complete the increasing map between fundamental intervals, and extend equivariantly along bump iterates. If both bump supports lie in one fundamental interval of another positive motion $h$, fix that cell's endpoints and repeat the conjugator on all its $h$-translates. The result commutes with $h$, and with any automorphism supported in the interval fixed pointwise.