Some Studies On The Strictly Partial Transformation Semigroup APT_n,k

Let Xn={1,2,…,n} if nature number n≥3.An,Sn,Tn and Pn are respectively alternating group,symmetric group,full transformation semigroup and partial transformation semigroup on Xn.Let Qk={α∈An:(?)x∈{k+1,k+2,…,n}(?)xα=x},APTn,k=Qk∪{Pn\Tn},then APTn,k be the subsemigroups of partial transformation Pn.And let Singn=Tn\Sn,SPn=Pn\Sn,then the Singn and the SPn are respectively called the Singular transformation semigroup and the partial Singular transformation semigroup on Xn.We say that a transformationα is order-preserving if x≤y implies xα≤yα,for all x,y∈dom(α).Let On={α∈Singn:for any x,y∈Xn,x≤y(?)xα≤yα} and let POn={α∈SPn:for any x,y∈dom(α),x ≤y(?)xα≤yα},the On and the POn are respectively called the order-preserving transformation semigroup and the partial order-preserving transformation semigroup on Xn.Let SPOn=POn\On,SPOnk={α∈SPOn:(?)x∈dom(α),x≤k(?)xα≤k}.For n≥3,0≤r≤n-2,let V(n,r)={α∈Pn\Tn:|im(α)|≤r},APTn(n,r)=Qn∪V(n,r).Then both the semigroup APTn(n,r)and the semigroup SPOnk are subsemigroups of semigroup APTn,k.This paper takes semigroup APTn,k,semigroup APTn(n,r)and semigroup SPOnk as the research objects,and the study found that the relevant results of the rank of semigroup APTn,k and the classification of the H-type maximal subsemigroup,and the rank and the completely classification of the maximal subsemigroup of the semigroup APTn(n,r);and characterize the Green(*)relations of the semigroups SPOnk and prove that semigroups of SPOnk is a non-regular abundant semigroup.