Orderability of the prefix expansion of an ordered inverse semigroup

We answer two orderability questions about the prefix expansion semigroup Pr(G) of an inverse semigroup G. We show that if G is a left ordered inverse semigroup, then Pr(G) is a left ordered inverse semigroup if and only if it is an ordered inverse semigroup, if and only if G is a semilattice. We also prove that when G and Pr(G) are left ordered, Pr(G) is proper if and only if G is proper. Positivity of the canonical map from G into Pr(G) is also proved. At the end we correct an existing result in the literature by showing that for two arbitrary inverse semigroups G and H the map Pr( \(\pi \) $\pi$ ): Pr(G) \(\longrightarrow \) $⟶$ Pr(H) induced by the partial homomorphism \(\pi \) $\pi$ : G \(\longrightarrow \) $⟶$ H is not necessarily a homomorphism, but is a partial homomorphism.