0-primitive ordered permutation groups

Abstract

Let G be a transitive ^-subgroup of the lattice-ordered group A{) of all order-preserving permutations of a chain . (In fact, many of the results are generalized to partially ordered sets and transitive groups G such that < implies g = for some positive geG, thus encompassing some results on non-ordered permutation groups.) The orbits of any stabilizer subgroup G a , ae, are convex and thus can be totally ordered in a natural way. The usual pairing <-> f ={ag I aeg} establishes an o-anti-isomorphism between the set of "positive" orbits and the set of "negative" orbits. If is an o-block (convex block) of G for which G a = , then ' is also an o-block. If G a has a greatest orbit 9 then { e I ' < < } constitutes an o-block of G. A correspondence is established between the centralizer Z A {)G and a certain subset of the fixed points of G a . The main theorem states that every o-primitive group ((?, ) which is not o-2-transitive or regular looks strikingly like the only previously known example, in which is the reals and G = {fe A() \( + )f=f+l for all e }. The "configuration" of orbits of G a must consist of a set o-isomorphic to the integers of "long" (infinite) orbits with some fixed points interspersed; and there must be a "period" ZQZ A ()G ( the Dedekind completion of ) analogous to the map z + 1 in the example. Periodic groups are shown to be ^-simple, and more examples of them are constructed.