【作者】 李艳；

【导师】 喻秉钧；

【作者基本信息】 四川师范大学 ， 基础数学， 2007， 硕士

【摘要】 在本文中,弱逆半群的结构定理第一次得到了完整的刻画:设S°是逆半群,其幂等元半格双序集为E°; E是弱逆双序集, EP = {e∈E : (?)f∈E, S(f,e) (?)ω(e)}是E的半格双序子集;θ是从EP到E°的双序同构;φ是从EP到对称弱逆半群PT (E∪S°)的映射.若四元组(S°,E,θ,φ)满足六条公理,我们可以构作对称弱逆半群PT (S°)的一个弱逆子半群Σ,其主元逆子半群与S°同构,其幂等元双序集与E(双序)同构.上述φ称为弱逆映射, (S°,E,θ,φ)称为弱逆系,Σ称为(S°,E,θ,φ)的弱逆包.反之,给定一个弱逆半群S,记S°= I(S), E°= E(S°)为逆半群S°的幂等元半格双序集, E = E(S)是S的幂等元双序集,则E是弱逆双序集, E_P = E°.对任意g∈E,定义g°是L-类L_g~S中惟一的主幂等元.进而,定义φ: E_P→PT (E∪S°)为:(？)e∈EP, domφe =∪{R_f∈E/←→: f∈L_e},(?)g∈Rf, gφe∈VP(f)∩Re_e~S°∩L_g°~S°.那么, (S°,E,1_E°,φ)是弱逆系,它的弱逆包Σ是与S同构的弱逆半群.在此基础上,我们给出了以上结构定理的同构定理,即刻画了两个弱逆包同构的充要条件,并给出了一个反例对这个充要条件加以说明.最后利用我们的结构定理对印度学者S. Madhavan在1980和1988年提出的两类双单弱逆半群的结构给出了一个更加简洁的刻画.更多还原

【Abstract】 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.更多还原