Hahn group ($V(\Gamma ,\mathbb{R})$)

For a totally ordered set $\Gamma$, take real coefficient families whose support satisfies the maximum condition: every nonempty subset of the support has a largest index. Add coefficients pointwise and decide the sign from the largest nonzero coordinate. This defines an ordered Abelian group. For $\Gamma ={\mathbb{Z}}_{>0}$ with its usual order, the maximum condition forces finite support. Some presentations reverse the index order and use a minimum instead; specify the convention when applying an embedding theorem.

Let a divisible ordered Abelian group be the union of a strictly increasing sequence of convex subgroups $C_1=0,C_2,\dots$, with each ordered quotient $C_{n+1}/C_n$ isomorphic to the additive real line. Convexity makes every $C_n$ divisible, so these are rational vector subspaces. Choose a rational-linear section for each quotient. Every element then has a unique finite sum of section values. Its sign is the sign of the coefficient of the largest active index, because convexity makes the corresponding ordered quotient decide its sign. The coefficients give an embedding, indeed an ordered-group isomorphism, into the Hahn group with positive integer indices in the maximum-support convention. No compatibility of a separate real scalar multiplication is needed for this group embedding.