Some remarks on almost \(l\)-groups (Q999099)

The divisibility group of every Bézout domain is an abelian \(l\)-group. Conversely, Jaffard, Kaplansky, and Ohm proved that each abelian \(l\)-group can be obtained in this way, which generalizes Krull's theorem for abelian linearly ordered groups. \textit{T. Dumitrescu, Y. Lequain, J. L. Mott}, and \textit{M. Zafrullah} [``Almost GCD domains of finite \(t\)-character'', J. Algebra 245, No. 1, 161--181 (2001; Zbl 1094.13537)] proved that an integral domain is almost GCD if and only if its divisibility group is an almost \(l\)-group. Then they asked whether the Krull-Jaffard-Kaplansky-Ohm theorem on \(l\)-groups can be extended to the framework of almost \(l\)-groups, and asked under what conditions an almost \(l\)-group is lattice-ordered. They asked the following questions: Question 1.1. Can every almost \(l\)-group be regarded as the group of divisibility of an almost GCD domain? Question 1.2. Under what conditions is an almost \(l\)-group an \(l\)-group? In this note the author answers these two questions.