The Order-reversing Quasi-idempotents Of Order-preserving And Order-reversing Transformation Semigroups

Let [n] = {1, 2,... ,n} ordered in the standard way, we denote by Tn the semigroup of all transformation of [n]. Let α ∈Tn, we say that a transformation α in Tn is order-preserving if , for all x,y E [n],x ≤y（?）xα≤yα（include identity transformation 1）; on the contrary, if ,for all x,y E [n],x < y（?）xα≥yα, we say that a transformation α in Tn is order-reserving. We denote by ODn the subsemigroup of Tn of all order-preserving and order-reserving transformations, call it the order-preserving and order-reserving transformation semigroup. An element α of Tn is called idempotent if α2 =α and quasi-idempotent if α2≠α, but α4 =α2. Because the transformations of ODn just have the order-preserving and order-reserving transformations, so we can put it into two categories: order-preserving quasi-idempotent and order-reserving quasi-idempotent.The following results of are given:In Chapter 2, we study the order-reserving quasi-idempotent rank of the semigroup ODn, the following results are given:Corollary 2.5 Let n > 2, OVn = (ξ1,ξ2,…ξn-1-h>, where ξi∈H(i,i₁[i]，1≤i≤ n—1. That it to say ODn can be generated of order-reserving quasi-idempotentWhere ξ∈ RQE（Jn-1）,1≤i≤n-1.Theorem 2.7 Let n ≥2, let where α1 > a2 >…> an-1; A1 < A2 < …< An-1; leti,i + 1 ∈ Ai is only a single point set, the α is the order-reserving quasi-idempotent if and only if there exists unique α∈H（i,j+l）[i]∪H（i,i+1）[n-i+1]∪H（n-i,n-i+1）[i]∪H（n-i,n-i+l）[n-i+1]1≤i≤n-Theorem 2.11 Let n > 1, then the order-reserving quasi-idempotent rank of semi-group ODn is n. That is to say rqdrank（ODn） = n.In Chapter 3, we study the maximal order-reserving quasi-idempotent generated subsemigroups of the semigroup ODn,Corollary 3.2 Let 1 < r < n - 1, then Tr = (RQE（Jr）>.Corollary 3.5 Let S is maximal order-reserving quasi-idempotent generated sub-semigroups of the semigroup ODn, then each one of the following types is a maximal order-reserving quasi-idempotent generated subsemigroups of the semigroup ODn（A） Ln-1;（B） Ln-2 ∪ M（n,i）∪ Jn, 1≤i≤n-1.In Chapter 4, we study the maximal regular order-reserving quasi-idempotent gen-erated subsemigroups of the semigroup ODn,Theorem 4.1 Let n≥3, if Sif the maximal regular order-reserving quasi-idempotent generated subsemigroups of the semigroup ODn, then each of the following types is a max-imal regular order-reserving quasi-idempotent generated subsemigroups of ODn（A） S1=Ln-1.（B） S2=Ln-2∪(h∪G\G（1）>.（C） S3=Ln-2∪<h∪G\{G（i）,G（n-i+1）}>,where 1<i≤p,p=[n/2].（D） S4=Ln-2∪(h∪G\{G（i）,G（n-i）}>, where 1≤i≤p,p=[n/2].Theorem 4.3 ODn is regular semigroup.