Definition of "abelian group"

Let G and H be partially ordered abelian groups. A positive homomorphism from G to H is any abelian group homomorphism f : G → H {\displaystyle f:G\rightarrow H} that maps positive elements to positive elements, that is, f ( G + ) ⊆ H + {\displaystyle f(G^{+})\subseteq H^{+}} .

1986, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 2010 softcover reprint, page 12

Chapter 2 is a brief introduction to some fundamental techniques for countable torsion-free abelian groups.

2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, Softcover reprint of 1st edition, page 74

By the end of Chapter 2 we had the full power of the Pontryagin Duality Theorem for compact abelian groups and for discrete abelian groups. Locally compact abelian groups are much closer to compact abelian groups than is apparent at first sight.

2013, Karl H. Hofmann, Sidney A. Morris, The Structure of Compact Groups: A Primer for the Student: A Handbook for the Expert, Walter de Gruyter, page 299