Some remarks on almost l-groups

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. Dumitrescu, Lequain, Mott, and Zafrullah [3] 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 [3, Questions 1 and 2]. This note answers the two questions.