Let G be a transitive
l-subgroup of the lattice-ordered group A(Ω) of all order-preserving permutations of a
chain Q. (In fact, many of the results are generalized to partially ordered sets Ω and
transitive groups G such that β < γ implies βg = γ for some positive g ∈ G,
thus encompassing some results on non-ordered permutation groups.) The
orbits of any stabilizer subgroup Gα,α ∈ Ω, are convex and thus can be
totally ordered in a natural way. The usual pairing Δ ←→ Δ′ = {αg|α ∈ Δg}
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
ΔGα= Δ, then Δ′ is also an o-block. If Gα has a greatest orbit Γ, then
{β ∈ Ω|Γ′ < β < Γ} constitutes an o-block of G. A correspondence is established
between the centralizer ZA(Ω)G and a certain subset of the fixed points of
Gα.
The main theorem states that every o-primitive group (G,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 = {f ∈ A(Ω)|(β + 1)f = βf + 1 for all β ∈ Ω}. The
“configuration” of orbits of Gα 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” z ∈ ZA(Ω)G (Ω the Dedekind completion of Ω) analogous to the map βz =β + 1 in the example. Periodic groups are shown to be l-simple, and more examples of
them are constructed.
