Vol. 55, No. 2, 1974

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Monotonic permutations of chains

Thomas Jerome Scott

Vol. 55 (1974), No. 2, 583–594

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 Kis also a lattice and there exist orps k in Ksuch that k2 = 1. The stabilizer of a point α Ω is Mα = {m Mαm = α}, and the paired orbit of Δ is Δ= {αgα 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.

Mathematical Subject Classification
Primary: 06A55
Received: 12 March 1974
Published: 1 December 1974
Thomas Jerome Scott