Partial orders on Dlab groups (Q2746914)

For every subgroup \(H\) of rank \(1\) in a multiplicative group of positive reals, complete descriptions are presented for maximal partial orders and for minimal isolated partial orders on the following Dlab groups: \(D_{H}({\mathbb I})\), \(D_{H*}({\mathbb I})\), \(D_{*H}({\mathbb I})\), and \({\overline{D}}_{H}({\mathbb I})\) of the unit interval \({\mathbb I}=[0,1]\) and \(D_{H}\) and \(D_{H*}\) of the extended real line \(\mathbb{\overline{R}}\). It is worth noting that \textit{W.~Ch.~Holland} [in: Algebra, Proc. Int. Conf. Memory A. I. Mal'tsev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 1, 197-207 (1992; Zbl 0766.06015)] gave a description for minimal and maximal partial orders of the group \(A(\mathbb R)\) of all order automorphisms of the linearly ordered set \(\mathbb R\) of reals.