MP-equivalence of free paratopological groups

Let FP(X) denote the free paratopological group over a topological space X. Two topological spaces X and Y are called MP-equivalent if FP(X) and FP(Y) are topologically isomorphic. At first, it is shown that there exist non-homeomorphic topological spaces X and Y such that FP(X) and FP(Y) are topologically isomorphic. Secondly, MP-invariance of free paratopological groups is investigated. It is established that pseudocompactness, hereditary Lindelofness, hereditary separability and the property of being a cosmic space are all MP-invariant, which generalizes some conclusions valid for free topological groups to free paratopological groups. Finally, a few questions about MP-equivalence of free paratopological groups are posed. (C) 2016 Elsevier B.V. All rights reserved.