The Research Of Monotone And K-preserving Transformations Semigroups

Let[n]=?<2<···<n} ordered in the standard way,and Jn be the finite full transformation semigroup under composition transformations from[n]to[n].A trans-formation a in Jn is called order-preserving transformation if x?y implies that xa?ya for all x,y?[n].A transformation a in Jn is called an order-reversing transformation if x?y implies that xa>ya for all x,y?[n].If On is a subsemigroup of Tn consist-ing of all order-preserving transformations on[n],we say that On is a order-preserving transformations semigroup.Let Rn be a order-reversing set of Tn and let OR,=On U Rn.Then OR is a subsemigroup of Tn consisting of all order-preserving transformations or order-reversing transformations on[n],we say that ORn is a monotone transformation semigroup.Suppose k?[n],let On(k)={??On:k??k},Rn(k)={{??Rn:k?=k},ORn(k)=On(k)?Rn(k),we say that OR,(k)is monotone and k-preserving transformations semigroups.In this paper,the following results of are given:In Chapter 1,introduces concepts related to semigroups.In Chapter 2,we study the regularity and Green relation of the semigroup ORn(k),the results of are given:Theorem 2.1 Letn?4,1 ?k?n,then ORn(k)is regular semigroup.Theorem 2.2.2 Letn>4,1 ?k?n,if?,??ORn(k),then(1)?L?(?)im(?)=im(?),(2)?R?(?)ker(?)=ker(?),(3)?D?(?)?and? Is the same type oraand ?Is the alien.In Chapter 3,we study the rank of the semigroup ORn(k),the following results are given:Theorem 3.17 Let n>4,then when n is even,rank(ORn(k))=When n is odd,rank(ORn(k))=(?)In Chapter 4,we will consider combinatorial results for semigroups of ORn(k):Theorem 4.3.Let 1?k?n,then |ORn(k)|=(k=1 2k-2)(n=k 2n-2k)+(k=1 n-1)2-1.