On reversing the order of a lattice ordered group
If G is a lattice-ordered group, let \(G_ R\) and \(G_ W\) be the lattice-ordered groups obtained by reversing the order and the multiplication \((a*b=ba)\) respectively. Then \(g\mapsto g^{-1}\) is an isomorphism between \(G_ R\) and \(G_ W\). Hence if V is a variety of lattice-ordered groups, then \(V_ R=\{G_ R:\quad G\in V\}=\{G_ W:\quad G\in V\}=V_ W.\) Moreover, the defining identities for \(V_ R\) are obtained from those for V by interchanging all meets and joins. The map \(V\mapsto V_ R\) is shown to be a lattice semigroup automorphism of the lattice-ordered semigroup of all varieties of lattice-ordered groups. If \(F=\{V:\quad V_ R=V\},\) then F includes all varieties defined by group laws and both F and its complement are shown to have size the continuum. Certain favourite varieties are examined to see if they belong to F or its complement; in particular, F includes all varieties that contain the variety of lattice-ordered groups generated by totally ordered groups. The techniques are elementary and quite computational.
scientific article; zbMATH DE number 3863261