| In the numerous branches of the study of semigroup, the research on transformation semigroup is most important in the theory of the semigroup algebraic. The reasonable explanation lies in its theoretical research value and widespread application. In the recent dozens years, many scholars have attached great importance to the research of transformation semigroup and substructure, and most important findings have been obtained. In this paper,we consider a class subsemigroups of singular transformation semigroup.Let Xn be finite set with n elements, Singn be semigroup consisting of all singular transformations on Xn, and A be an arbitrary nonempty subset of Xn . DefineThen S(Xn, A) is a subsemigroup of Sinyn. In this paper,wc mainly consider that (generalize)the Green's relations and idempotent-generated property, and we also obtain some combinatorial results of semigroup S(Xn, A).In chapter 1, we present some basic definitions and symbols, and proof that S(Xn, A) (?) S(Xn, B) for arbitrary A, B (?) Xnand | A |=| B |.In chapter 2, we study the Green's relations of S(Xn,A), and obtain that equivalent character for L,R,D relation in the semigroup S(Xn,A) and especially in some regular D-classes.In chapter 3, we discuss the Green's star relations of S(Xn, A), and obtainequivalent character of L*,R*, D* relation are obtained. It is shown that S(Xn,A)(1＜| A |< n) is a non-regular abundant semigroup with n - 1 D*-classes and D* = (?).In chapter 4,we firstly consider property of digraph associated E(Jn-1*).By introducing definition of maximal strong-complete in paper [33]of You T.J. ,it is obtained that the direction digraph associated E(Jn-1*) is maximal strong-complete.Further,elements that are generated by idempotents are defined in Jn-1*.Let FA = {α∈Re* : e∈I1,and aα= a,(?)∈A}, FXn\A = {α∈Re* : e∈I2,and aα= a,(?)∈Xn\A}, T = {α∈Re* : e∈I2,and A (?) imα}. it is testified that only the three type; dements can be expressed as product of idempotents in Jn-1*.In chapter 5, we consider combinatorial results of some subsets of S(Xn, A) (| A |= k＞1) .Let S(I1∪I2) be a set consisting of all idempotent-generated elements in Jn-1* and Si = {α∈S(Xn,A) :| imα∩A |= i}(i = 1,2). It is shown that... |
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