Weakly least integer closed groups (Q2568682)

This paper can be considered as a continuation of a previous article by the same authors [in: J. Martínez (ed.), Ordered algebraic structures, Kluwer, Dordrecht, Develop. Math. 7, 245--260 (2002; Zbl 1074.06005)] where least integer closed groups were dealt with. Now the authors investigate a generalization of the mentioned notion; they consider weakly least integer closed groups (\(\mathbf{wLIC}\) groups, for short). Let \(\mathbf W\) be the class of all archimedean lattice-ordered groups \(A\) with a distinguished weak unit \(e_A\). In the whole paper, \(A\) is identified with the canonical Yosida representation as an \(\ell\)-group of extended real-valued functions on a certain compact Hausdorff space \(YA\). For a real \(r\), let \([r]\) be the least integer greater than or equal to \(r\). If \(t\in\{ +\infty, -\infty\}\), let \([t]=t\). If \(A\in\mathbf W\) and \(a\in A\), then the element \([a]\) of \(A\) is defined component-wise. An element \(A\in \mathbf W\) is defined to be a \(\mathbf{wLIC}\) group if for each \(a\in A\), there is \(a'\in A\) for which \([a]=a'\) on some dense subset of \(YA\). Sample results: An intrinsic characterization of \(\mathbf{wLIC}\) groups is presented. It is proved that \(\mathbf{wLIC}\) is a hull class. The relation between projectivity, weak projectivity and the property of being a \(\mathbf{wLIC}\) group are studied in detail. A series of well-chosen examples is given.