Topological monomorphisms between free paratopological groups (Q690859)

Let \(Y\) be a Tychonov topological space. The free paratopological group \(FP(Y)\) and the abelian free paratopological group \(AP(Y)\) over \(Y\) were introduced in [\textit{S. Romaguera}, \textit{M. Sanchis} and \textit{M. G. Tkachenko}, Topol. Proc. 27, No. 2, 613-640 (2003; Zbl 1062.22005)] in analogy to the notions of free topological group \(F(Y)\) and abelian free topological group \(A(Y)\) over \(Y\). Let \(X\) be a subspace of \(Y\) and let \(e_{X,Y}:X\to Y\) denote the embedding map; then the extension \(\hat{e}_{X,Y}:AP(X)\to AP(Y)\) is a continuous monomorphism. The main result of this paper characterizes the case when \(\hat{e}_{X,Y}:AP(X)\to AP(Y)\) is a topological monomorphism. This is the counterpart for abelian free paratopological groups of a known result for abelian free topological groups proved in [\textit{M.G. Tkachenko}, Sov. Math., Dokl. 27, 341--345 (1983); translation from Dokl. Akad. Nauk SSSR 269, 299--303 (1983; Zbl 0521.22002)]. An analogous characterization for free topological groups is given in [\textit{E. Nummela}, Topology Appl. 13, 77-83 (1982; Zbl 0471.22001)] and [\textit{V.G. Pestov}, Vestn. Mosk. Univ., Ser. I 1982, No. 1, 35--37 (1982; Zbl 0499.22001)] when \(X\) is dense in \(Y\). So the same problem is considered here for free paratopological groups, and in case \(X\) is dense in \(Y\) a necessary condition is found to have that \(\hat{e}_{X,Y}:FP(X)\to FP(Y)\) is a topological monomorphism. The theory of quasi-uniformities and new results about quasi-pseudometrics on (abelian) free paratopological groups are applied to prove these theorems.