Topological monomorphisms between free paratopological groups

Suppose that X is a subspace of a Tychonoff space Y. Then the embedding mapping e(X,Y) : X -> Y can be extended to a continuous monomorphism (e) over cap (X,Y) : AP(X) -> A P(Y), where A P(X) and A P(Y) are the free Abelian paratopological groups over X and Y, respectively, in this paper, we mainly discuss when (e) over cap (X,Y) is a topological monomorphism, that is, when (e) over cap (X,Y) is a topological embedding of AP(X) to AP(Y).