This paper is a sequal to
o-primitive ordered permutation groups [Pacific J. Math., 40 (1972), S49-372]. There
it was shown that if A(Ω) is the lattice-ordered group of all o-permutations of a chain
Ω, and if G is an l-subgroup of A(Ω) which is periodically o-primitive (transitive and
lacking proper convex blocks, but neither o-2-transitive nor regular), then the
(convex) orbits of any stabilizer subgroup Gα,α ∈ Ω, themselves form a chain
o-isomorphic to the integers. Let Δ be any nonsingleton orbit of Gα. Here it is shown
that Gα is faithful on Δ and that Gα|Δ is o-2-transitive and contains an
element ≠1 of bounded support. From this it follows that all o-primitive groups
(except for certain pathological o-2-transitive groups) are complete l-subgroups
of A(Ω), and hence are completely distributive. When G is “full”, Gα|Δ
satisfies an important “splice” property, and Gα and G are laterally complete.
There is a detailed description of the unique full group G for which Δ is an
α-set, and a lisling of the other “nice” permutation group representations of
G.
