Lateral completion and structure sheaf of an Archimedean \(l\)-group (Q952253)

The notion of lateral completion of a lattice-ordered group has been investigated by several authors. Let \(\widetilde{G}\) be the structure sheaf of an abelian lattice-ordered group \(G\). The set of all prime \(\ell\)-ideals of \(G\) endowed with the natural topology is denoted by Spec\(^*G\). The authors define \(E(G)\) to be the direct limit \(\varinjlim\Gamma(U,\widetilde{G})\), where \(U\) runs through the dense open subsets of Spec\(^*G\). The main result of the paper consists in proving that if \(G\) is Archimedean then \(E(G)\) coincides with the lateral completion of \(G\). This result is then applied for constructing the essential closure of \(G\).