| Let natural number n≥3,Xn={1,2,…,n｝.In,Sn and An are partial one-to-one transformation semigroup,symmetric group and alternating group on a finite set Xn respectively.Let SIn=In\Sn,SIn is subsemigroup of In obviously,SIn is defined as part-one singular transformation semigroup.Let Qk={α∈An:?x∈{k+1,…,n} has xα=x} be k-local alternating group on Xn if for an arbitrary integer k such that 3≤k≤n,let AIn,k=Qk U SIn.It is easy to prove that AIn,k is subsemigroup of the partial one-to-one transformation semigroup In.Let Z be a ring of integer,define a relation on the Z×Z:T(Z×Z)+={(a,b)|a,b ∈ Z,a≤b} and an operation ■ on the T(Z×Z)+,for all(a,b),(c,d)E T(Z×Z)+,(a,b)■(c,d)=(a+c,b+d).It is easy to prove that(T(Z×Z)+,■)is a semigroup.The details are as follows:Chapter 1 Introduction and preliminaryChapter 2 Starting with Green’s relations and regularity of semigroup Aln,k,generative relationship and ideals of semigroup AIn,k are characterized,furthermore,the rank and the cubic idempotent rank range of semigroup AIn,k are determined.Based on the rank and the cubic idempotent rank of semigroup AIn,k,the completely classifications of Gmaximal subsemigroups and isolated subsemigroups of semigroup AIn,k are proved.Chapter 3 By studying the elements of the semigroup T(Z×Z)+,we describe its Green’s relations,regularity and rank.Chapter 4 Summary and prospect... |
