Morita Theory For Fair(Po)Semigroups

In this dissertation,we mainly investigate Morita theory for fair(po)semigroups.We characterize Morita-like equivalence for fair semigroups and discuss the relation of Morita equivalence of fair posemigroups;finally,we study Morita invariants of posemigroups with weak local units.The dissertation is divided into four chapters.The main results are given as follows:In the first chapter,we give the introduction and preliminaries.In the second chapter,we mainly investigate Morita-like equivalence and Mori-ta context for right fair semigroups.If two right fair semigroups S and T are Morita-like equivalence,that is,there is an category equivalent F:US-Aet(?)UT-Act:G,we characterize the two functors F and G using Hom functor and tense product functor.Also,we investigate Morita context for right fair semigroups and obtain equivalence between two right unitary act categories.The main results are given as follows:Theorem 2.1.G Let S and T be two semigroups.Suppose(S,T,SPT,TQS,?)is a Morita context,where:P(?)T Q?S and ?:Q(?)S P?T are biact mor-phisms such that ImT?U(SS)and Im?—U(TT).Then we have the following natural functor isomorphisms:-(?)SP?Homs(QS,-)U(Tt),-(?)TQ?HomT(PT,-)U(SS)Theorem 2.1.8 Let S and T be two right fair semigroups such that U(SS)and U(TT)have common weak right local units.Assume S and T are Morita-like equivalence via F:US-Act(?)UT-Act:G.Let P=F(U(Ss))and Q=G(U(TT)).Then(1)PT and Qs are respectively generators in UT-Act and US-Act.(2)End(PT)U(SS)?U(Ss)and End(Qs)U(TT)?U(TT)as semigroups.(3)SPT and TQS are unitary biacts.(4)SPT?Homs(QS,U(SS))U(TT)and TQS?HomT(PT,U(TT))U(SS).(5)F ? Uoms(Qs,-)U(TT)and G?HomT(PT,-)U(SS).(6)SPT and TQS induce a Morita context(S,T,TQS,?,?)such that ? and? are biact homorphisms such that Im?=U(SS)and Im?-U(TT),respectively.Moreoer.if we define multiplications in Q(?)s P and P(?)T Q,respectively:(y(?),x)(y'(?)x')=y(?)?(x(?)y')x\(x(?)y)(x'(?)y')=x(?)?(y(?)x')y',where.x,x' ? P and y,y'? Q,then Q(?)s P and P(?)T Q are semigroups,and ? and? are semigroup homomorphisms.(7)F?-(?)S P and G ?-(?)TQ.Theorem 2.2.3 Let S and T be two right fair semigroups such that U(S)and U(T)have common weak right local units.Assume(S,T,S PT,TQs,?.?)is a Morita context,where SPT and TQs are strong s-unital and Im?=U(Ss)and Im?=U(TT).We have the following conditions:(1)QS and PT are respectively generators of US-Act and UT-Act.(2)P(?)T Q??SU(SS)S and Q(?)SP??TU(TT)T as biacts.Furthermore,if we define multiplications in P(?)T Q and Q(?)S P respectively by:(x(?)y)(x'(?)y')y')=x(?)?(y(?)x')y',(y(?)x)(y'(?)x')=y(?)?(x(?)y')x',where x,x' ? P and y,y'?Q,then P(?)? Q and Q(?)s P are semigroupsi and? and?are semigroups isomorphisms.(3)sPT?Homs(Qs,U(SS))U(TT)and TQSs?HomT(PT,U(TT))U(SS)as biacts.(4)U(TT)?End(Qs)U(TT)and U(SS)?End(PT)U(SS)as semigroups.(5)The functor pair(-(?)SP,-(?)TQ)defines an equivalence-(?)sP:US-Act=UT-Act:-StQ-That is.S and T are Morita-like equivalence semigroups.(6)The functor pair(Homs(Cs,-)U(TT),HomT(PT,-)U(SS)defines an e-quivalence Homs(Qs,-)U(TT):US-Act(?)UT-Act:HomT(PT,-)U(PT,-)U(Ss).That is,S and T are Morita-like equivalence semigroups.(7)The lattice of right ideals of U(Ss)(resp..U(TT))is isomorphic to the lattice of subacts of PT(resp.,QR).Furthermore,these induce lattice isomorphisms between the lattices of two-sided ideals of U(Ss)(resp.,U(TT))and the lattice of subacts of SPT(resp.,TQS).In the third chapter,we mainly investigate Morita equivalence of right fair posemigroups whose unitary part have common weak right local units,and obtain equivalent characterizations of Morita equivalence of these posemigroups.The main results are given as follows:Theorem 3.2.6 Let S be a right fair posemigroup with WRDP such that.I=U(SS)has common weak right local units.Then(1)the categories Pos-US and Pos-UI are equivalent;(2)the categories Pos-FS and Pos-FI are equivalent.Theorem 3.2.8 Let S,T be two right fair posemigroups with WRDP such that U(Ss),U(TT)have common weak right local units.Then the followings are equivalent.(1)The categories Pos-US and Pos-UT are equivalent.(2)The categories Pos-FS and Pos-FT are equivalent.(3)The posemigroups S and T are Morita equivalent.(4)The categories Pos-UU(SS)and Pos-UU(TT)are equivalent.(5)The categories Pos-FU(Ss)and Pos-FU(TT)are equivalent.(6)The posemigroups U(SS)and U(TT)are Morita equivalent.In the fourth chapter,we mainly investigate Morita invaria.nts of posemigroups with weak local units.The semigroups in this chapter have weak local units.We get some properties of prime subposets,semiprime subposets,m-systems,n-systems,strongly convex kernels,strongly convex semiprimary subposets and obtain some results on these invariants under Morita equivalence of posemigroups with weak local units.The main results are given as follows:Theorem 4.2.5 Let S and T be two strongly Morita equivalent posemigroups.Then the following statements are true:(1)M is a strongly convex prime subposet of P if and only if for strongly convex subposets A and B of P.g2(A)B=Ag3(B)? M implies A ? M or B ? M.(2)N is a strongly convex semiprime subposet of P if and only if for any strongly convex subposet A of P.g1(A)A=Ag3(A)(?)N implies A(?)V.Theorem 4.2.8 Let S and T be two strongly Morita equivalent posemigroups.Then the following statements hold:(1)For all p,p'? H,there is an element q?Q such that ?(p(?)q)p'=p?(q(?)p')?H,then H is an m-svstem of P.(2)For all p?H,there is an element q?Q such that ?(p(?)q)p=p?(q(?)p)?H,then H is an n-svstem of P.Theorem 4.2.9 Let S and T be two strongly Morita equivalent posemigroups.Then the following statements hold:(1)There exists an inclusion preserving bijection between the set of all prime ideals of S and the set of all strongly convex prime subposets of P.(2)There exists an inclusion preserving bijection between the set of all semiprime ideals of S and the set of all strongly convex semiprime subposets of P.Theorem 4.3.7 Let S and T be two strongly Morita equivalent posemi-groups.Then there exists an inclusion preserving bijection between the set of all semiprimary ideals of S and the set of all strongly convex semiprimary subposets of P.