Free paratopological groups

Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a Pα-space X where α is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a neighborhood base at the identity of the group 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 FP(X) is T0 if X is T0, we characterize the spaces X for which the group FP(X) is a topological group and then we give a class of spaces X for which the group FP(X) has the inductive limit property.