Vol. 49, No. 2, 1973

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The lattice-ordered group of automorphisms of an α-set

Stephen H. McCleary

Vol. 49 (1973), No. 2, 417–424

The group of all automorphisms of a chain Ω forms a lattice-ordered group A(Ω) under the pointwise order. It is well known that if G is the symmetric group on elements (6), then every automorphism of G is inner. Here it is shown that if Ω is an α-set, every l-automorphism of A(Ω) (preserving also the lattice structure) is inner. This is accomplished by means of an investigation of the orbits ωA(Ω) of Dedekind cuts ω of Ω.

The same conjecture for arbitrary chains Ω has been investigated in [6], [4], and [8]. Lloyd proved in [6] that when Ω is the chain of rational numbers (i.e., the 0-set), or is Dedekind complete, every l-automorphism of A(Ω) is inner. He also stated this conclusion for α-sets in general, but a lacuna in his proof has been pointed out by C. Holland.

Mathematical Subject Classification
Primary: 06A55
Received: 24 May 1973
Published: 1 December 1973
Stephen H. McCleary