The continuous $d$-open homomorphism images and subgroups of $mathbb{R}$-factorizabile paratopological groups

In this paper, we prove that: (1) Let $f:Grightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhangs result cite[Theorem 1.7]{PZ} is improved. (2) Let $G$ be a regular $mathbb{R}$-factorizable paratopological group; then every subgroup $H$ of $G$ is $mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. This result gives out a positive answer to an question of M.~Sanchis and M.~Tkachenko cite[Problem 5.3]{ST}.