Some Properties On Two Kinds Of Subsemigroups Of Partial Transformation Semigroup

Let the natural numbers n≥3,the finite set Xn=1,2,…,n},Pn,Sn,An and POn are respectively partial transformation semigroup,symmetric group,alternating group and partial order-preserving transformation semigroup on Xn.SPn=Pn\Sn be singular partial transformation semigroup on Xn.Let Qk={α∈An:(?)x∈{k+1,…,n},xα=x} be k-local alternating group on Xn if for an arbitrary integer k such that 3≤k≤n and let APn,k=Qk∪SPn.Let POn(k)={α∈POn:(?)x∈dom(α),x≤k(?)xα≤k if for an arbitrary integer k such that 1≤k≤n.It is easy to prove that APn,k and POn(k)are subsemigroups of the partial transformation semigroup Pn.In this paper,some properties of the semigroup APn,k and the semigroup POn(k)are studied,the details are as follows:By analyzing the characteristics of elements with rank r in the semigroup APn,k,the Green-relation,regularity,generating relation and ideal of semigroup APn,k are described.The minimal generating set and the minimal cubic idempotent elements generating set of the semigroup APn,k are determined,and then the rank and the cubic idempotent rank of the semigroup APn,k are obtained.Furthermore,the structure and classification of the maximal(regular)subsemigroups of the semigroup APn,nr are considered.In addition,by studying the characteristics of elements of the semigroup POn(k),the Green(star)-relation,regularity and abundance are described.