Studies Of Quasi-idepotent Of Order-preserving Transformation Semigroup

Let[n]={1,2,...,n} ordered in the standard way.We denote by Singn the semigroup (under composition) of all singular tranformations of[n].We say that a transformation α in Singn is order preserving if,for all x,y∈[n],x≤y implies xα≤yα.We denote by On the subsemigroup of Singn of all order preserving singular transformations.An element a of any given semigroup is called idempotent if a2≠a and quasi-idempotent if a2≠a and a4=a2.In this paper,we mainly study the quasi-idempotents of the semigroup On.The following results are given:In Chapter 2,we study the quasi-idempotent of the semigroup On of rank n-1:Theorem 2.3 Let n≥3.Let wherea1<a2<…<an-1,A1<A2<…<An-1,then α is quasi-idempotent if and only if there exists unique k∈{1,…,n-1} such that ak(?)Ak.In Chapter 3,we study the ideal of the semigroup OnTheorem 3.1 Let n≥3,let E2(Jn-1)be the set of all quasi-idempotents of Jn-1, then E2(Jn-1)={αi,i+1:1≤i≤n-2}∪{αi+1,i+2:1≤i≤n-2}.Theorem 3.4 Let n≥3,let E2(Jn-1)be the set of all quasi-idempotents of Jn-1, then On=<E2(Jn-1)>.Theorem 3.8 Let n≥5,let E2(Jn-2)be the set of all quasi-idempotents of Jn-2, then O(n,n-2)=<E2(Jn-2)>.In Chapter 4,we study the quasi-idempotent-generated maximal idempotent-generated subsemigroup of the semigroup On:Theorem 4.1 Let n≥6,then the quasi-idempotent-generated maximal idempotent-generated subsemigroup of the semigroup On have the following forms:(1)Ak=O(n,n-2)∪(Jn-1\Lk),k=1,n(2)Bk=O(n,n-2)∪(∪i=1kLi)∪(∪i=k+1n-1R(i,i+1)),k=4,…,n-3.(3)Ck=O(n,n-2)∪(∪i=1kR(i,i+1))∪(∪i=k+2nLi),k=3,…,n-4.In Chapter 5,we study the maximal quasi-generated subsemigroup of OnTheorem 5.1 Let n≥6. Let S be the quasi-idempotent-generated maximal idempotent-generated subsemigroup of the semigroup On,then S is a maximal quasi-generated subsemigroup of On.