Factoriality in Riesz groups

Throughout let G = (G,+,≤, 0) denote a Riesz group, where + is not necessarily a commutative operation. Call x ∈ G homogeneous if x > 0 and for all h, k ∈ (0, x] there is t ∈ (0, x] such that t ≤ h, k. In this paper we develop a theory of factoriality in Riesz groups based on the fact that if x ≤ G and x is a finite sum of homogeneous elements then x is uniquely expressible as a sum of finitely many mutually disjoint homogeneous elements. We then compare our work with existing results in lattice-ordered groups and in (commutative) integral domains.