Prime radicals of \(AO\)-groups (Q950892)

Let \(G\) be a partially ordered group, two positive elements \(a\) and \(b\) of \(G\) are almost orthogonal if \(c \leq a\) and \(c \leq b\) imply \(c^n \leq a\) and \(c^n \leq b\) for any natural number \(n\) and for all \(c\in G\). A directed group \(G\) is an \(AO\)-group if each \(g \in G\) can be expressed as \(g=ab^{-1}\), where \(a\) and \(b\) are almost orthogonal elements in \(G\). There are several radicals for lattice-ordered groups in the literature. In the paper under review, the authors define and discuss prime radicals of \(AO\)-groups. Some interesting results on commutants are also obtained.