一类变换半群的秩

On the rank of a kind of transformation semigroups

设TX为集合X上的全变换半群,E是X上一个等价关系.令TE(X)={f∈TX: (x,y)∈E,(f,x),f(y))∈E},则TE(X)是TX的一个子半群.本文讨论对于一个较为特殊的情况,即E只有两个等价类,且每个等价类有n(n≥3)个点.结果发现,这时TE(X)有一组生成元,含有5个元素,从而确定了TE(X)的秩不超过5.

Let TX be the full transformation semigroup on the set X, E be an equivalence on X, letTE(X)={f∈TX:(x,y)∈E,(f(x),f(y))∈E}.Then TE(X) is a subsemigroup of TX.In this paper,a special case is considered,that is,the equivalence E has two classes each of which is of n points.It is found that TE(X) has a generating set containing 5 elements.Then it is determined that the rank of TE(X) is no more than 5.