Orderability of the prefix expansion of an ordered inverse semigroup

Abstract

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(\(\pi \)): Pr(G) \(\longrightarrow \) Pr(H) induced by the partial homomorphism \(\pi \): G \(\longrightarrow \) H is not necessarily a homomorphism, but is a partial homomorphism.