Lexico groups and direct products of lattice ordered groups (Q2810994)

The paper under review studies lexico groups. A lattice-ordered group \(A\) is a lexico group if there exists a convex \(\ell \)-subgroup \(A_0\) of \(A\) such that for each \(a \in A\setminus A_0\) we have either \(a <a_0\) or \(a_0<a\) for each \(a_0 \in A_0\). The main result asserts the following: Let \(A\) be a convex \(\ell \)-subgroup of a lattice-ordered group \(G\) such that (i) \(A\) is a lexico group, and (ii) \(A\) fails to be upper bounded in \(G\). Then, \(A\) is a direct factor of \(G\).