半群Q(k)的极大正则子半带

Maximal Regular Subsemiband of Semigroup Q(k)

设T(X)和O(X)分别是X上的全变换半群和保序全变换半群,Y是X的非空子集,令F(X,Y)={α∈T(X):Xα?Yα?Y},OF(X,Y)=O(X)∩F(X,Y).当Y=n≥4时,对任意的2≤k≤n-2,考虑半群Q(k)={α∈OF(X,Y):Im(α)≤k}的极大正则子半带的结构,利用Miller-Clifford定理,证明了半群Q(k)的极大正则子半带有且仅有两类:A(α)=Q(k-1)∪(J(k)\L_α),α∈J(k);B(β)=Q(k-1)∪(J(k)\R_β),β∈N(k).

Let T(X)and O(X)be the full transformation semigroup,full order-preserving transformation semigroup on X,respectively,and Y be a nonempty subset of X,F(X,Y)={α∈T(X):Xα?Yα?Y},OF(X,Y)=O(X)∩F(X,Y).When Y =n≥4,for an arbitrary integer k such that 2≤k≤n-2,we considered the structures of maximal regular subsemibands of semigroup Q(k).Using Miller-Clifford theorem,we proved that Q(k)had exactly two classes of maximal regular subsemibands,i.e:A(α)=Q(k-1)∪(J(k)\L_α),α∈J(k);B(β)=Q(k-1)∪(J(k)\R_β),β∈N(k).