The Structure Of Weakly Inverse Semigroups

In this thesis, the structure of weakly inverse semigroups isfirstly characterized: let S°be an inverse semigroup with semilattice biorder setof idempotents E°; E a weakly inverse biordered set with a semilattice biordersubset EP = {e∈E : (?)f∈E, S(f,e) (?)ω(e)} with a biorder-isomorphismθfrom EP onto E°. Moreover, there is a mappingφfrom EP into the symmet-ric weakly inverse semigroup PT (E∪S°) such that the quadruple (S°,E,θ,φ)satisfies six appropriate conditions, then a weakly inverse semigroupΣcan beconstructed in PT (S°) with I(Σ)～= S°, E(Σ) E. The mappingφis called aweakly inverse mapping relating E to S°, (S°,E,θ,φ) is called a weakly inversesystem and theΣis called the weakly inverse hull of (S°,E,θ,φ).Conversely, given a weakly inverse semigroup S, denoting S°= I(S), E°=E(S°) the idempotent semilattice biordered set of the inverse semigroup S°,E = E(S) the biordered set of idempotents of S, then E is a weakly inversebiordered set with EP = E°. For any g∈E, denoting by g°the unique principalidempotent in the L-class L_g~S , defineφ: E_P→PT (E∪S°) as follows:?e∈EP, domφe =∪{Rf∈E/←→: f∈L_e},(?)g∈Rf, gφe∈VP(f)∩R_e~S°∩L_g°~S°.Then, (S°,E,1E°,φ) is a weakly inverse system whose weakly inverse hullΣis aweakly inverse semigroup isomorphic to S.Furthermore, an isomorphism theorem for this structure theorem is givenwhich characterizes the necessary and su?cient condition for two weakly inversehulls to be isomorphic. Meanwhile a counterexample is provided to illustrate thecondition. Finally by using our structure theorem, new characterizations of bisimpleweakly inverse semigroups with partial identities or partial right unitoids aregiven, which are more concise than those given by Indian scholar S Madhavan in1980 and 1988 respectively.