Search

We explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of a Galois groupoid. Finally, we prove that in the category of preordered groups there is also a pretorsion theory, whose torsion subcategory can be identified with a category of internal groups. This latter is precisely the subcategory of protomodular objects in the category of preordered groups, as recently discovered by Clementino, Martins-Ferreira, and Montoli. [ABSTRACT FROM AUTHOR]

Using congruences, a Hoehnke radical can be defined for graphs in the same way as for universal algebras. This leads in a natural way to the connectednesses and disconnectednesses (= radical and semisimple classes) of graphs. It thus makes sense to talk about ideal-hereditary Hoehnke radicals for graphs (= hereditary torsion theories). All such radicals for the category of undirected graphs which allow loops are explicitly determined. Moreover, in contrast to what is the case for the well-known algebraic categories, it is shown here that such radicals for graphs need not be Kurosh–Amitsur radicals. [ABSTRACT FROM AUTHOR]

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions ( C) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions ( C) or ( C). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]

We first characterize τ-complemented modules with relative (pre)covers. We also introduce an extending module relative to τ-pure submodules on a hereditary torsion theory τ and give its relationship with τ-complemented modules. [ABSTRACT FROM AUTHOR]

A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3-for-2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory. [ABSTRACT FROM AUTHOR]