The lattice-ordered group of automorphisms of anα-set
Stephen H. McCleary · Pacific Journal of Mathematics · 1973
THEOREM 1 (Lloyd).If Ω is o-2-homogeneous, then every l-automorphism of A(Ω) is inner, provided that no orbit ώA(Ω), ώ e Ω\Ω, is oisomorphic to Ω.It may be that the proviso that no orbit ώA(Ω), ώ e Ω\Ω, be o-isomorphic to Ω is satisfied by every o-2-homogeneous Ω; this is an 417 418 STEPHEN H. McCLEARY open question. 1 We shall find at any rate that the proviso holds when Ω is an α-set.For any ώ e Ω y Ω o-2-homogeneous, the orbit ώA(Ω) is dense in Ω.For g e A(Ω), form g e A(ώA(Ω)) by first extending g to Ω and then restricting to ώA(Ω).The map g ->g is an ^-isomorphism of A(Ω) into A{ώA(Ω)).We shall write (A(Ω), ώA(Ω)) when considering A(Ω) to act on ώA(Ω), and shall say that (A(Ω), ώA(Ω)) is entire if the Z-isomorphism is onto A(ώA(Ω)).PROPOSITION 2. Suppose that A{Ω) is o-2-transitive on Ω, and let ώeΩ\Ω.Then A(Ω) is also o-2-transitive on ώA(Ω).