Radical subgroups of lattice ordered groups.

*(English)*Zbl 0605.06013Let G be an \(\ell\)-group and c(G) the lattice of all convex \(\ell\)- subgroups of G. A lattice ordered group \(H\in c(G)\) will be said to be a radical subgroup of G if, whenever \(G_ 1\in c(G)\) and \(H_ 1\in c(H)\) such that \(G_ 1\) is isomorphic to \(H_ 1\), then \(G_ 1\subseteq H\). The system R(G) of all radical subgroups of G is partially ordered by inclusion. If G has no nontrivial radical subgroups, then G is said to be r-homogeneous. G will be said to be totally r-inhomogeneous if, whenever \(0\neq H\in R(G)\), then there exists \(H_ 1\in R(G)\) such that \(\{0\}\subset H_ 1\subset H.\)

The main results of this paper concern the lattice R(G) for the case when G is a complete \(\ell\)-group. Let us mention the following existence results: For each \(\alpha >0\) there exists a proper class \(A_{\alpha}\) of mutually nonisomorphic complete \(\ell\)-groups such that for each \(G\in A_{\alpha}\), R(G) is isomorphic to the Boolean algebra \(2^{\alpha}\). For each ordinal \(\delta\) there is a complete \(\ell\)-group G such that R(G) is a chain isomorphic to \(\delta\). For each complete \(\ell\)-group G there exists a complete \(\ell\)-group \(G_ 1\) such that \(G\in R(G_ 1)\) and G is covered by \(G_ 1\) in the lattice \(R(G_ 1)\). There exists a proper class of mutually nonisomorphic totally r-inhomogeneous \(\ell\)- groups. The question whether there exists a complete totally r- inhomogeneous \(\ell\)-group \(G\neq \{0\}\) remains open. Some results on the lattice \({\mathcal R}_ c\) of all radical classes of complete \(\ell\)-groups are established.

The main results of this paper concern the lattice R(G) for the case when G is a complete \(\ell\)-group. Let us mention the following existence results: For each \(\alpha >0\) there exists a proper class \(A_{\alpha}\) of mutually nonisomorphic complete \(\ell\)-groups such that for each \(G\in A_{\alpha}\), R(G) is isomorphic to the Boolean algebra \(2^{\alpha}\). For each ordinal \(\delta\) there is a complete \(\ell\)-group G such that R(G) is a chain isomorphic to \(\delta\). For each complete \(\ell\)-group G there exists a complete \(\ell\)-group \(G_ 1\) such that \(G\in R(G_ 1)\) and G is covered by \(G_ 1\) in the lattice \(R(G_ 1)\). There exists a proper class of mutually nonisomorphic totally r-inhomogeneous \(\ell\)- groups. The question whether there exists a complete totally r- inhomogeneous \(\ell\)-group \(G\neq \{0\}\) remains open. Some results on the lattice \({\mathcal R}_ c\) of all radical classes of complete \(\ell\)-groups are established.

Reviewer: F.Šik

##### MSC:

06F15 | Ordered groups |

20F60 | Ordered groups (group-theoretic aspects) |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

06B15 | Representation theory of lattices |

##### Keywords:

lattice of convex \(\ell \)-subgroups; lattice ordered group; radical subgroups; complete \(\ell \)-groups; totally r-inhomogeneous \(\ell \)- groups; radical classes of complete \(\ell \)-groups
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