0-primitive ordered permutation groups
Stephen H. McCleary · Pacific Journal of Mathematics · 1972
Let G be a transitive ^-subgroup of the lattice-ordered group A{Ω) of all order-preserving permutations of a chain Ω.(In fact, many of the results are generalized to partially ordered sets Ω and transitive groups G such that β Δ f ={ag I aeΔg} establishes an o-anti-isomorphism between the set of "positive" orbits and the set of "negative" orbits.If Δ is an o-block (convex block) of G for which ΔG a = Δ, then Δ' is also an o-block.If G a has a greatest orbit Γ 9 then {β e Ω I Γ' < β < Γ} constitutes an o-block of G.A correspondence is established between the centralizer Z A {Ω)G and a certain subset of the fixed points of G a .The main theorem states that every o-primitive group ((?, Ω) which is not o-2-transitive or regular looks strikingly like the only previously known example, in which Ω is the reals and G = {fe A(Ω) \(β + ϊ)f=βf+l for all β e Ω}.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 (Ω)G (Ω the Dedekind completion of Ω) analogous to the map βzβ + 1 in the example.Periodic groups are shown to be ^-simple, and more examples of them are constructed.