On convex linearly ordered subgroups of an \(h\ell \)-group (Q2777497)

Let \(G\) be a right partially ordered group. Let us denote by \(G\uparrow \) (and \(G\downarrow \)) the set of all \(x\in G\) such that for all \(y,z\in G\) the implication \(y\leq z \Rightarrow xy\leq xz\) (and \(xy\geq xz\), respectively) is valid. Then \(G\) is called a half lattice-ordered group (an \(h\ell \)-group) if the partial order on \(G\) is not trivial, \(G={G\uparrow}\cup {G\downarrow}\) , and \({G\uparrow}\) is a lattice. Then \(G\uparrow \) is a lattice-ordered group (an \(\ell \)-group) and a normal subgroup of \(G\). Let \(\mathcal H_1\) be the class of all \(h\ell\)-groups which are not \(\ell \)-groups. The notion of an \(h\ell\)-group was introduced by \textit{M. Giraudet} and \textit{F. Lucas} [Fundam. Math. 139, 75-89 (1991; Zbl 0766.06014)], where it is also proved that if \(G\in \mathcal H_1\), and if the \(\ell \)-group \(G\uparrow \) is linearly ordered, then \(G\uparrow \) is abelian. Moreover, \(G\uparrow \) is in this case a maximal convex linearly ordered subgroup of \(G\). The authors of the paper, among others, generalize this result proving the following theorem: Let \(G\in \mathcal H_1\) and let \(X\) be a maximal convex linearly ordered subgroup of \(G\). If there exists \(a\in G\) such that \(e\neq a\), \(a^2=e\) and \(aX=Xa\), then \(X\) is an abelian \(\ell \)-group.