Study On Special Subsemigroups Of The Semigroup AT_n,k

Let T_n,S_n,A_n,P_n and O_n be full transformation semigroup、symmetry group、alternating group、partial transformation semigroup and orderpreserving transformation semigroup on a finite chain X_n if natural number n≥3,respectively.Let Q_k={α∈A_n:（?）x∈{k+1,k+2,…,n1,n},xα=x} be the k-local alternating group on X_n and let AT_n,k=Q_k∪（T_n\S_n）if for an arbitrary integer k such that 1 ≤k≤n.It is easy to prove that semigroup AT_n,k is subsemigroup of the full transformation semigroup T_n.Let O_n^[l,m]={α∈O_n:（?）x∈X_n,l≤xα≤<m}if for an arbitrary integer l,m such that l,m∈X_n.It is easy to prove that semigroup O_n^[l,m] is subsemigroup of the semigroup O_n and AT_n,k,and O_n^[1,n]=O_n.In this paper,special properties of the semigroup AT_n,k and the semigroup O_n^[l,m]are described.The details are as follows:By studying the elemental characteristics and egg box graph of semigroup AT_n,k,the Green-relation,regularity,generating relation and ideal of semigroup AT_n,k are characterized.Analyzing the element of the semigroup AT_n,k with rank r,and then the rank and the cubic idempotent rank of the semigroup AT_n,k are determined.Further,the isolated subsemigroups of semigroup AT_n,n and maximal（regular）subsemigroups of semigroup AT_n,n^r are considered when k=n.Finally,by studying the elemental characteristics and egg box graph of semigroup O_n^[l,m],the Green（star）relation and regularity of semigroup O_n^[l,m] are obtained.Further,non-regular abundance be determined.