保持等价关系的变换半群的组合结果
A Combinatorial Result for Certain Semigroups of Transformations Preserving an Equivalence Relation

$设$ \mathscr{T}_X $是非空集合X上的全变换半群，E是X上的（n，m）-型等价关系，则 $ {T_E}\left( X \right) = \left\{ {f \in {\mathscr{T}_X }:\forall x, y \in X, \left( {x, y} \right) \in E \Rightarrow \left( {f\left( x \right), f\left( y \right)} \right) \in E} \right\} $ 是$ \mathscr{T}_X $的子半群.计算了变换半群T_E（X）的基数，并且在n=2，m ≥ 2和n=3，m ≥ 2的条件下，分别给出了T_E（X）的正则元个数的计算公式.$
$Let $\mathscr{T}_X $ be a full transformation semigroup on the nonempty set X, and E be an (n, m)-type equivalence relation on X. Then $ {\mathscr T_E}\left( X \right) = \left\{ {f \in {\mathscr{T}_X }:\forall x, y \in X, \left( {x, y} \right) \in E \Rightarrow \left( {f\left( x \right), f\left( y \right)} \right) \in E} \right\} $ is a subsemigroup of T_X. In this paper, we calculate the cardinality of the semigroup T_E(X) and present a formula for the number of its regular elements under the supposition that n=2, m ≥ 2 and n=3, m ≥ 2.