TITLE:
The Lattice of Fully Invariant Subgroups of the Cotorsion Hull
AUTHORS:
Tariel Kemoklidze
KEYWORDS:
Lattice of Fully Invariant Subgroups; Direct Sum of Torsion-Complete Groups; Cotorsion Hull
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.8,
November
27,
2013
ABSTRACT: The paper considers the lattice of fully
invariant subgroups of the cotorsion hull when a separable primary group Tis an arbitrary direct sum of torsion-complete
groups.The investigation of this problem in the case of a cotorsion hull is
important because endomorphisms in this class of groups are completely defined
by their action on the torsion part and for mixed groups the ring of
endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if
and only if the group is a fully invariant subgroup of the cotorsion hull of
its torsion part. In the considered case, the cotorsion hull is not fully
transitive and hence it is necessary to introduce a new function which differs
from an indicator and assigns an infinite matrix to each element of the
cotorsion hull. The relation difined on the set of these matrices is different from the
relation proposed by the autor in the countable case and better discribes the
lower semilattice. The use of the relation essentially simplifies the verification of the
required properties. It is proved that the lattice of fully invariant subgroups
of the group is isomorphic to the lattice of filters of the
lower semilattice.