A Class Of Orthodox Transformation Semigroups

Let X2n={1,2,…,2n} be a finite chain,T2n be the full transformation semigroup on X2n.Denoted by DM(2n,n)=DO(2n,n)U DR(2n,n),whereDO(2n,n)={α∈T2n:((?)x,y∈X2n)|xα-n|≤|yα-n|(?)|x-n|≤|y-n|},DR(2n,n)={α∈T2n:((?)x,y∈X2n)|xα-n|≥|yα-n|(?)|x-n|≤|y-n|}.LetODM(2n,n)={α∈DM(2n,n):((?)i∈{1,2,…,n-1})(n-i)α≤(n+i)α},Obviously,ODM(2n,n)is a subsemigroup of T2n.This paper mainly discusses some prop-erties of the semigroup ODM(2n,n):In chapter 2,we study the regularity and green’s relation of the semigroup ODM(2n,n).The following results are obtained:Theorem 2.1.2 The semigroup ODM(2n,n)is a regular subsemigroup of T2n.Theorem 2.1.4 Let n≥3,then ODM(2n,n)be an orthodox transformation semi-group.Theorem 2.2.4 Let α,β ∈ ODM(2n,n),then(1)(α,β)∈L if and only if Im(α)=Im(β).(2)(α,β)∈R if and only if ker(α)=ker(β).Theorem 2.2.7 Let α,β∈ODM(2n,n),then(α,β)∈D if and only if |Im(α)|=|Im(β)| and one of the following holds.(1)Im(α)=Im(β);(2)ker(α)=ker(β);(3)When Im(α)≠Im(β),ker(α)≠ker(β),ker(α)is complemental to ker(β).In chapter 3,we study the rank of ODM(2n,n)and E(ODM(2n,n)),The following results are obtained:Theorem 3.1.9 Let n≥ 3,then rank(ODM(2n,n))=[n/2]×2+1.Theorem 3.2.6 Let n≥3,then rank(E(ODM(2n,n)))=2n-1.In chapter 4,we study the maximal subsemigroups of the semigroup ODM(2n,n).The following results are obtained:Theorem 4.5 Let n≥3,and n is odd,then the maximal subsemigroups of ODM(2n,n)are of the following three types:(1)Mη=ODM(2n,n)\{η};(2)Mi-=ODM(2n,n)\{α(Ai,i),α(An-i,n-i)};(3)Mi+=ODM(2n,n)\{α(Ai,2n-i),α(An-i,n+i)}.Theorem 4.6 Let n≥3,and n is even,then the maximal subsemigroups of ODM(2n,n)are of the following the following six types:(1)Mη=ODM(2n,n)\{η};(2)Mi-=ODM(2n,n)\{α(Ai,i),α(An-i,n-i)};(3)Mi+=ODM(2n,n)\{α(Ai,2n-i),α(An-i,n+i)};(4)Mn/2-=ODM(2n,n)\{α(An/2,n/2),e(An/2,n/2)};(5)Mn/2+=ODM(2n,n)\{α(An/2,3n/2),e(An/2,3n/2)};(6)Mn/2=ODM(2n,n)\{α(An/2,n/2),α(An/2,3n/2)}.