2012/01/24 by Elijah Stines, Stines, Elijah
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1201.5081
openalex publication_date 2012/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.