Consider a short exact sequence of partially ordered groups
$$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a convex subgroups of $G.$ In content of ordered groups how do we define the spliting condition of this sequence? Suppose if this sequnce splits, then there exists a homomorphism $\gamma : G/H \rightarrow G$ such that $\beta\circ\gamma$ is identity on $G/H.$ Do we need $\gamma$ is an order homomorphism?



Spliting of short exact exact sequences of partially ordered groups
Let $G$ be a (Abelian) po-group and remember that the partial order is completely determined by the positive cone $G^+=\{g\in G:g\geq 0\}$. Furthermore, a subset $P\subseteq G$ is a positive cone if and only if it has the following properties:


-- $0 \in P$;


-- if $a \in P$ and $b \in P$, then $a+b \in P$;


-- if $a \in P$ and $-a \in P$ then $a=0$.


Now, given a subgroup $H$ of $G$, $H$ is a subobject of $G$ if and only if $H^+=G^+\cap H$. On the other hand, in the category of po-groups, kernels of morphisms and subobjects are two different concepts as, if we want $H$ to be a kernel, then we need also to assume that it is convex in $G$, that is, if $h\_1\leq g\leq h\_2$ for $h\_1,h\_2\in H$ and $g\in G$, then $g\in H$.


Now, in the question (and comments), it seems to be asked about torsion-free Abelian groups, the torsion free hypothesis adds one more difficulty, as the kernels in such category are the pure subgroups. Thus, to start with the answer we need to assume that $H$ is a pure, convex subgroup of $G$, whose order is just the restriction of that of $G$. In this case, $G/H$ is a torsion free Abelian po-group when we take as positive cone $(G/H)^+=\{g+H:g\in G^+\}$.


Let me also recall that, in the category of po-groups, the coproduct of two po-groups $(G\_1, G\_1^+)$ and $(G\_2,G\_2^+)$ is given by $(G\_1\oplus G\_2, G\_1^+\oplus G\_2^+)$.


Now, in the setting of the question, even if $\alpha$ is a pure monomorphism, and there exists an ordered homomorphism $\gamma:G/H\to G$ such that $\beta\gamma=id\_{G/H}$ we just obtain that $G\cong H\oplus G/H$ as abstract groups because there is a priori no reason for $G$ to split as a po-group. To ensure this stronger splitting one need to check that $G^+=\alpha(H^+)\oplus \gamma((G/H)^+)$.


