Free paratopological groups. II
Let FPG(X) and FP(X) be the free paratopological groups on a topological space X in the senses of Graev and Markov, respectively. In this paper, we prove that the groups FPG(X) and FP(X) are discrete if X is discrete and the group FPG(X) is indiscrete if X is indiscrete while the group FP(X) is the union of infinite indiscrete subspaces if X is indiscrete. Then we give a class of spaces X for which the groups FP G(X) and FP(X) are locally invariant and another class of spaces X where they are not. Finally, we provide another proof for the existence of the free paratopological group FP(X) by embedding the space X in an infinite direct product of paratopological groups. 2013 Topology Proceedings.