Abelian Quasi-Ordered Groups

Details Abelian Quasi-Ordered Groups Abelian Quasi-Ordered Groups

2012-01-24

arxiv.org/abs/1201.5081v1

Mathematics

Rings and Algebras

Scientific paper

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.

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