The Complete Classification Of The Maximal Subsemigroups Of Finite Order-preserving Transformation Semigroup O_n

In[1],S is the the maximal subsemigroups of O_n,professor Yang has obtained the complete classification of the maximal subsemigroups of the finite order-preserving transformation semigroups O_n when | E(J_n-1\ S)|＜3.In this paper,we give the the complete classification of the maximal subsemigroups of On when | E(J_n-1\ S)|≥3.First,we give four notations DD_2r+1,UU_2r+1,DU_2r+1and UD_2r+1. (1＜k₁＜k₂＜…＜k_2r+2＜n-1,k_2i-1＞k_2i,k_2i+1＜k_2i+1,i = 1,2,…,r + 1)Second,we prove all of them are the subsemigroups of O_n.Third,we are going to testify that they are all the maximal subsemigroups of On by the properties we have given.Forth,we verify the following is correct.If S is a maximal subsemigroup of O_n,|E(J_n-1\ S)|= n≥3.When n = 2r + 1,then E(J_n-1\S)has only two idempotent chains.(â… ).E(J_n-1\S)= {[k₁+1→k₁],[k₂→k₂+1],…,[k_2r→k_2r+1],[k_2r+1+1→k_2r+1]},(1≤k₁＜k₂＜…k_2r+1＜n-1,k_2i-1+1＜k_2i,k_2i＜k_2i+1,i =1,2,…,r)(â…¡).E(J_n-1\S)= {[k₁→k₁ + 1],[k₂ + 1→k₂],…,[k_2r+ 1→k_2r],[k_2r+1→k_2r+1+ 1]},(1＜k₁＜k₂＜…＜k_2r+1＜n,k_2i-1＜k_2i,k_2i+ 1 < k_2i+1,i = 1,2,…,r)When n = 2r + 2,then E(J_n-1\S)has only two idempotent chains.(â…¢).E(J_n-1\S)= {[k₁ + 1→k₁],[k-2→k₂+1],…,[k_2r+1+ 1→k_2r+1],[k_2r+2→k_2r+2+ 1]},(1≤k₁≤k₂＜…2r+1＜n,k_2i-1+1＜k_2i,k_2i＜k_2i+1,i= 1,2,…,r).(â…£).E(J_n-1\S)= {[k₁→k₁ + 1],[k₂+1→k₂],…,[k_2r-1→k_2r+1+1],[k_2r+2+ 1→k_2r+2]},(1＜k₁＜k₂＜…＜k_2r+1＜n-1,k_2i-1＜k_2i,k_2i+ 1 < k_2i+1,i = 1,2,…,r).Fifth,we demonstrate that in(â… ),S = DD_2r+1;in(â…¡),S = UU_2r+1;in(â…¢),S= DU_2r+2;in(â…£),S = UD_2r+2.Thus,we have given the complete classification of the maximal subsemigroups of the finite order-preserving transformation semigroup O_n when | E(J_n-1\S)[≥3.