An automorphism (opp) of a
chain Ω is a permutation g of Ω which preserves order in the sense that ω < τ iff
ωg < τg. All anti-automorphism (orp) is a permutation k of Ω which reverses order in
the sense that ω < τ iff ωk > τk. A permutation which either preserves or reverses
order is called monotonic, and the group of all monotonic permutations is denoted by
M(Ω). M(Ω) is ordered pointwise, i.e., g ≦ h iff ωg ≦ ωh for all ω ∈ Ω. This yields a
po-set but not a po-group. However the subgroup A(Ω) of all opps of Ω forms a
lattice-ordered group (l-group). A subgroup K of M(Ω) is called l-monotonic if
K′ = K ∩ A(Ω) is nonempty, i.e., if K contains an orp, and if G(K) = K ∩ A(Ω) is
a transitive l-subgroup of A(Ω). The group M(Ω) is l-monotonic iff Ω is
homogeneous and admits an orp. The opp group G(K) has index 2 in K and is
o-isomorphic to K′. Thus K′ is also a lattice and there exist orps k in K′ such
that k2= 1. The stabilizer of a point α ∈ Ω is Mα= {m ∈ M∣αm = α},
and the paired orbit of Δ is Δ′ = {αg∣α ∈ Δg for some g ∈ G}. The Main
Theorem 8 shows that a Kα-orbit is the union of a Gα-orbit and its paired
Gα-orbit.
An l-subgroup H of A(Ω) is extendable if there exists an l-monotonic group
(K,Ω) such that G(K) = H. Regular abelian opp groups and full periodically
o-primitive groups are uniquely extendable. There exist both extendable and
nonextendable o-2-transitive groups. A characterization of o-primitive l-monotonic
groups is given.
