Free paratopological groups

Let $\text{FP}(X)$ be the free paratopological group on a topological space $X$ in the sense of Markov. In this paper, we study the group $\text{FP}(X)$ on a $P_{\alpha }$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group $\text{FP}(X)$ is an Alexandroff space if $X$ is an Alexandroff space. Moreover, we introduce a neighborhood base at the identity of the group $\text{FP}(X)$ when the space $X$ is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group $\text{FP}(X)$ is $T_0$ if $X$ is $T_0$, we characterize the spaces $X$ for which the group $\text{FP}(X)$ is a topological group and then we give a class of spaces $X$ for which the group $\text{FP}(X)$ has the inductive limit property.