Some pathology involving pseudo $l$-groups as groups of divisibility

Abstract:

In a partially ordered abelian group $G$, two elements $a$ and $b$ are pseudo-disjoint if $a,b \geqq 0$ and either one is zero, or both are strictly positive and each $o$-ideal which is maximal with respect to not containing $a$ contains $b$, and vice versa. $G$ is a pseudo lattice-group if every element of $G$ can be written as a difference of pseudo-disjoint elements. We prove the following theorem: suppose $G$ is an abelian pseudo lattice-group; if there is an $x > 0$ and a finite set of pairwise pseudo-disjoint elements ${x_1},{x_2}, \cdots ,{x_k}$ all of which exceed $x$, and in addition this set is maximal with respect to the above properties, then $G$ is not a group of divisibility. The main consequence of this result is that every so-called “$v$-group” $V(\Lambda ,{R_\lambda })$ for a given partially ordered set $\Lambda$, and where ${R_\lambda }$ is a subgroup of the additive reals in their usual order, is a group of divisibility only if $\Lambda$ is a root system, and hence $V(\Lambda ,{R_\lambda })$ is a lattice-ordered group. We do give examples of pseudo lattice-groups which are not lattice-groups, and yet are groups of divisibility. Finally, we compute for each integral domain $D$ whose group of divisibility is a lattice-group, the group of divisibility of the polynomial ring $D[x]$ in one variable.