Topological monomorphisms between free paratopological groups

TL;DR

This paper investigates conditions under which the natural embedding of free Abelian paratopological groups over a subspace into those over a larger space is a topological embedding, extending classical results in topological algebra.

Contribution

It provides new criteria for when the extension of the embedding map between free Abelian paratopological groups is a topological monomorphism.

Findings

Characterizes when the embedding is a topological monomorphism.

Extends classical embedding results to paratopological groups.

Provides conditions for topological embedding in free Abelian paratopological groups.

Abstract

Suppose that $X$ is a subspace of a Tychonoff space $Y$. Then the embedding mapping $e_{X, Y}: X\rightarrow Y$ can be extended to a continuous monomorphism $\hat{e}_{X, Y}: AP(X)\rightarrow AP(Y)$, where $AP(X)$ and $AP(Y)$ are the free Abelian paratopological groups over $X$ and $Y$, respectively. In this paper, we mainly discuss when $\hat{e}_{X, Y}$ is a topological monomorphism, that is, when $\hat{e}_{X, Y}$ is a topological embedding of $AP(X)$ to $AP(Y)$.