A family of totally ordered groups with some special properties (Q2569336)

In this paper the following main results are obtained. a) It is proved that for any fixed cardinality \({\aleph}_k\) there exists a metrizable field \(K\) whose value group has cardinality \({\aleph}_k\). b) It is proved that for any fixed uncountable cardinal \({\aleph}_k\) there exists a metrizable field \(K\) of cardinality \({\aleph}_k\) which has an absolutely convex subset that is not countably generated as a \(B_K\)-module. c) It is proved that for any cardinality \({\aleph}_k > {\aleph}_0\) for the value group the two conditions (the whole group has a cofinal sequence and every subset of the group which is bounded above has a cofinal sequence) are logically independent.