The Study Of Transformation Semigroups CS_n

Set Xn={1,2,…,n} and give the order of natural numbers.Singn is a singular transfomation semigroup on Xn,Cn is cyclic group on Xn which is generated g=(12…n-1 n 23…n 1),Let CSn=Singn ∪ Cn,then semigroup CSn is a subsemigroup of Tn.In the first place,for any n≥ 3,we prove that the rank of CSn of semigroup is[n/2]+1.Secondly,on the basis of studying rank,we get the maximal subsemigroups of transformation semigroups CSn of complete classification.Finally,for any n≥3,we get the maximal regular subsemigroups of transformation semigroups CSn of complete classification.Furthermore,for any n≥4,and 3(?)n,we obtain the locally maximal idempotent correlation generating subsemigroups and locally maximal non-idempotent square idempotent correlation generating subsemigroups of transformation semigroups CSn.In this paper.The main results of the paper are given as follows:Theorem 2.6 Let n≥3,Then rankCSn=[n/2]+1.Theorem 3.9 Let n≥3,then the maximal subsemigroups of semigroup CSn have only the following forms:(1)Ot=<gft>∪ Singm,1≤t≤c,(2)Ok=Cn∪T(n,n-2)∪△n-1\[λ1k],2≤k≤[n/2]+1.Theorem 4.1.2 Let n≥3,then maximal regular subsemigroups of semigroup CSn is the same as the maximal subsemigroup form:(1)Ot=<gft>∪Singn,1≤t≤c,(2)Ok=Cn∪T(n,n-2)∪△n-1\[λ1k],2≤k≤[n/2]+1.Theorem 4.2.6 Let 3(?)n ≥ 4,and 2 ≤k≤[n2/]+1,then the structure of the locally maximal idempotent correlation generating subsemigroups of transformation semigroup CSn is as follows:Pk=Cn∪M∪T(n,,n-[n/2]-1).Theorem 4.3.3 Let 3(?)n≥4,and 2≤k≤[n/2]+1,then the structure of the locally maximal non-idempotent square idempotent correlation generating subsemigroups of transformation semigroups CSn:Pk=Cn∪M∪T(n,n-[n/2]-1).