A construction of weakly inverse semigroups

Let S ^o be an inverse semigroup with semilattice biordered set E ^o of idempotents and E a weakly inverse biordered set with a subsemilattice E _P = {e ∈ E | ∀ f ∈ E, S(f, e) ⊆ ω(e)} isomorphic to E ^o by θ: E _P → E ^o. In this paper, it is proved that if ∀f, g ∈ E, f ⟷ g ⟹ f ^o θ D ^S° g ^o θ and there exists a mapping ϕ from E _P into the symmetric weakly inverse semigroup P ℐ (E∪S ^o) satisfying six appropriate conditions, then a weakly inverse semigroup Σ can be constructed in P ℐ (S ^o), called the weakly inverse hull of a weakly inverse system (S ^o,E, θ, ϕ) with I(gS) ≅ S ^o, E(Σ) ≃ E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given. Supported by an NSF Grant of China #10471112