Abelian Quasi-Ordered Groups
Abstract
Abelian groups having partial orderings compatible with their binary operations have long been studied in the literature. In particular, lattice-ordered abelian groups constitute a universal-algebraic variety, and thus form a category which is monadic over the category of sets. The current paper studies the more general case of quasi-ordered abelian groups, identifying some of their more fundamental properties and their relationships to partially ordered and lattice-ordered groups. We reinterpret the category of quasi-ordered abelian groups with order preserving morphisms by examining the interplay between the group and the set of all positive elements under the quasi-ordering. The main result shows that the category of quasi-ordered abelian groups is monadic over the category of set monomorphisms.
Cite
@article{arxiv.1201.5081,
title = {Abelian Quasi-Ordered Groups},
author = {Elijah Stines},
journal= {arXiv preprint arXiv:1201.5081},
year = {2012}
}