Study On Two G-type Singular Transformation Semigroups

Let Xn=[n]={1,2,…,n} and give the order to natural numbers.Let Tn be the semigroup of full transformations on Xn,Sn be the symmetric group on Xn and Singn=Tn\Sn be the singular transformations semigroup on Xn.H(n,k),the local cyclic group with k on Xn,is generated byLet SH(n,k)=Singn U then SH(n,k)forms a subsemigroup of composition of trans-formation on Tn.Y(n-1),the cyclic group on Xn,is generated byLet SY(n-1)=Singn ∪ Y(n-1),then SY(n-1)forms a subsemigroup of composition of transformation on Tn.Two G-type singular transformation semigroups SH(n,k)and SY(n-1)are studied in this thesis.The contents are given in following:the rank of the transformation semigroup SH(n,k)and the maximal substructure of the transformation semigroup SH(n,n-1)for k=n-1 are studied.Let k≥ 2,n-k≥ 3,then rank SH(n,k)=(n-k)(n-k-1)/2+n-k+2+[k/2].The maximal subsemigroup of the semigroup SH(n,n-1)has the same form with the maximal regular subsemigroup and only have the following forms:let n>3,(1)Pt=<gt>∪ singn,the prime number t is in {2,3,…,n-1} and(t,n-1)≠1.(2)Pt1=H(n,n-1)∪T(n,n-2)∪△n-1\[e→1],2≤h≤[n-1/2]+1.(3)Pt2=H(n,n-1)∪T(n,n-2)∪Δn-1\[en→1].(4)Pt3=H(n,n-1)∪T(n,n-2)∪Δn-1\[e1→n].Meanwhile,the rank and the maximal subsemgroup of SY(n-1)are studied.The result of the rank is:let n≥ 5,then rankSY(n-1)=Cn-12+[n-1/2]/2+2+[n/2]the maximal subsemigroup only have the following forms:(1)Let i∈ {1,2,3,…,[n,2]},j∈{i+1,…,n-i-1,n-i,n}\(i+2},then Qij1=SY(n-1)\[[ei→j]is the maximal subsemigroup of the semigroup SY(n-1).(2)Let i∈{1,2,…,[n-2/2]},then Oi2=SY(n-1)\[βi]is the maximal subsemigroup of the semigroup SY(n-1).(3)(?)α∈ SY(n-1)and im(α)=Xn\{n},then QJ3=SY(n-1)\Lα is the maximal subsemigroup of the semigroup SY(n-1)、when n is odd;(?)a∈SY(n-1)and im(α)=Xn\{i},i={n/2,n}.then QO3=SY(n-1)\L is the maximal subsemigroup of the semigroup SY(n-1)when n is even.(4)Q4=G ∪ Singn is the maximal subsemigroup of the semigroup SY(n-1).