Reversional automorphisms of lattice-ordered groups (Q1921817)

A group automorphism \(\varphi\) of an \(\ell\)-group \(G\) is said to be reversional if \(x\leq y\) implies \(\varphi x\geq \varphi y\) for all \(x, y\in G\). Evidently, every finite-order reversional automorphism of an \(\ell\)-group has order 2. Since \(\varphi^2= 1\), it follows that \(\varphi= \varphi^{- 1}\). The authors prove the following: Let \(G\) be a totally ordered group and let \(A\) be a totally ordered Abelian group. The group \(\text{Awr}G\) with ordering of type (A) or (B) possesses no reversional automorphism. Therefore a totally ordered group is isomorphically embeddable into a totally ordered group without reversional automorphisms.