Abundancy For Semigroups Generated By Idempotent Transformations With Defect 1

Let Xn = {1,2,...,n}.The singular transformation semigroup Singn is a set of all singular transformations on the set Xn under the composition of transfor-mations.In this paper,the composition acts from the left to the right,i.e.let S be a transformation semigroup,(x)? o ?=(xa)? for all ?,?? S and x ? Xn.Let En-1 be the set of all the idempotent transformations with defect 1 in Singn,For any non-empty subset I of En-1,the subsemigroup S(I)generated by I is called the semigroup generated by idempotent transformations with defect 1.From 1966,Howie and many other researchers had established many results of the semigroup S(I),but no one has discussed the left and right abunda-ncy for S(I)in generally.Therefore investigating the Green*-relations on S(I)and its left and right abundancy shall be a natural and meaningful problem.We divide our paper into six chapters:Chapter 1:we introduce the development background of transformation semi-groups and present the research status of the semigroup S(I).Chapter 2:we present some basic concepts on the algebraic theory of semi-group and some established results and conclusions on the semigroup S(I).Chapter 3:we describe the C*-relation on S(I)and claim that these kind of semigroups are left abundant.Chapter 4:we are going to characterize the necessary and sufficient condition for I such that the following relation holds:(?,?)?R*(?)Ker(?)= Ker(?),(?)?,? S(I).Chapter 5:we put forward the concept of weakly partition order-decreasing singular semigroup and obtain a necessary and sufficient condition such that these kind of semigroups are abundant.Chapter 6:we summarise this paper and put forward the further research problems related to our topics.