The structure of free Abelian \(\ell\)-groups and free vector lattices
The paper gives the prime and regular subgroup structures for free abelian \(\ell\)-groups (lattice-ordered groups). An \(\ell\)-subgroup H (both a subgroup and a sublattice) of an \(\ell\)-group G is convex if \(0\leq g\leq h\in H\) forces \(g\in H\). A convex \(\ell\)-subgroup P is prime if \(a\wedge b=0\) forces either \(a\in P\) or \(b\in P\), and regular, if it is maximal with respect to not containing an element of G. If \(\mu\) is a cardinal, let \(A_{\mu}\) denote the free abelian \(\ell\)-group of rank \(\mu\). For an \(\ell\)-group G, the depth of G is defined to be the maximal cardinality of any chain of primes of G. \(\Gamma\) (G) denotes the set of the regular subgroups of G, partially ordered by inclusion. \(\Gamma\) (G) is then a root system: for any \(M\in \Gamma (G)\) the set \(\{\) \(N\in \Gamma (G)|\) \(M\subset N\}\) is totally ordered. Theorem 2.8. For a finite n, \(depth(A_ n)=n\). Thus every proper prime is regular. - Theorem 2.15. For finite \(n\geq 2\), \(\Gamma (A_ n)\) consists of countably infinitely many copies of \(\Gamma (A_{n- 1}{\overset \leftarrow \times}{\mathbb{Z}})\) and \(2^{\omega}\) many copies of \(\Gamma (A_{n-1})\). With minor adaptations, analogues for free vector lattices are proved. Some results are obtained in the infinite rank case.
scientific article; zbMATH DE number 4000001