The Study Of Partial Transformation Semigroups With N-Element Permutation

Let Xn＝{1,2,...,n} ordered in the standard way.Let Pn(Tn)be the semigroup of partial(full)transformations of the set Xn.Let Ln be the symmetric inverse semigroup on Xn.Let Cn(Sn)be the cyclic group(symmetric group)on Xn.Let Singn=Tn\Sn be the semigroup of all singular transformations of the set Xn,then Singn be the subsemigroup of Tn.Let SPn=Pn\Sn be the semigroup of all partial singular transformations of the set Xn,then SPn be the subsemigroup of Pn.Let SLn=Ln\Sn be the semigroup of all partial one to one transformations of the set Xn,then SLn be the subsemigroup of Ln.Let SCn=Cn ∪ Singn be the subsemigroup of Tn;let PSCn=Cn ∪ SPn be the subsemigroup of Pn;let LSCn=Cn ∪ SLn be the subsemigroup of Ln.In this thesis,we introduce the partial transformation semigroup with n-element permutation SCn,PSCn and LSC(n,r),in which study the rank and the structure of the maximal subsemigroup,the differences between the generating sets and the maximal subsemigroups of these special semigroups and common semigroups are obtained.The main results are given in follwing:In chapter 1,we give introduction and preliminaries.In chapter 2,we study the rank and the maximal subsemigroup of the semigroup SCn,the main results are given in follwing:Theorem 2.1.9 Let n>4,then rankSCn=Cn2+[n/2]/2+1.Theorem 2.2.1 Let i∈[1,[n/2]],j∈(i,n-i+1]\{i+2},then M1ij=SCn\[λji]is maximal subsemigroup of the semigroup SCn;Let i∈[1,[n-1/2]],then M2i=SCn\[αi]is maximal subsemigroup of the semigroup SCn.Theorem 2.2.2(?)α∈SCn,then M3=SCnn\Ca is maximal subsemigroup of the semigroup SCn,when n is odd and im(α)=Xn\{n+1./2}.Theorem 2.2.5 M4=g∪Singn is maximal subsemigroup of the semigroup SCn.In chapter 3,we study the rank and the maximal subsemigroup of the semigroup LSC(n,r),the main results are given in follwing:Theorem 3.1.10 Let n≥3,then rankLSC(n,r)=Cnr/2+3 when n is even and r is odd;in other case.rankLSC(n.r)=(?)+3.Theorem 3.2.2 Let n≥ 3,the set of all subset while cardinality is r of Xn denoted by Qr.a,2-partition of Qr denoted by(C1,C2),that C1 ∪ C2=Qr,C1 ∩ C2=(?).a 2-partition of C1 denoted by(C11,C12).Then only two R-classes are equivalent classes when n is even and r is odd by lemma 3.1.4,Let#12#12 Let#12#12#12#12 it has the following form:,is maximal subsemigroup of the semigroup LSC(n,r).Theorem 3.2.3 Let n≥ 3,the set of all subset while cardinality is r of Xn denoted by Qr.a 2-partition of Qr denoted by(C1,C2),that C1 ∪ C2=Qr,C1 C2=(?).a 2-partition of C1 denoted by(C11,C12),a 2-partition of C2 denoted by(C21,C22).Now have two R-classes are equivalent classes and one R-classes are equivalent classes when n is not even or r is not odd by lemma 3.1.4,Let#12#12#12 it has the following form:while BCijClk={α∈△r:domα∈Cij,ima ∈ Clk},{i,j,l,k}={1,2},is maximal subsemigroup of the semigroup LSC(n,r).Theorem 3.2.4 Let n≥ 3,then M7=G∪LS(n,r)is maximal subsemigroup of the semigroup LSC(n,r),while g is maximal subgroup of the cyclic group Cn.In chapter 4,we study the rank and the maximal subsemigroup of the semigroup PSCn,the main results are given in follwing:Theorem 4.1.8 Let n≥3,then rank(PSCn)=Cn2+3[n/2]/2+1.Theorem 4.2.1 Let i∈[1,[n/2]],j∈ {i,n-i+1]\{i+2} ｝,then M1ij=PSCn\[λji]is maximal subsemigroup of the semigroup PSCn;Let i∈[1,[n-1/2]],then M2i=PSCn\[αi]is maximal subsemigroup of the semigroup PSCn;Let 1≤h≤[n+1/2],then M3h=PSCn\[μhh]is maximal subsemigroup of the semigroup PSCn.Theorem 4.2.2 M4=g∪SPn is maximal subsemigroup of the semigroup PSCn,while g={g2} is maximal subgroup of the cyclic group Cn.