%0 Journal Article
%T 保持等价关系的变换半群的组合结果<br>A Combinatorial Result for Certain Semigroups of Transformations Preserving an Equivalence Relation
%A 孙垒
%J 西南大学学报(自然科学版)
%D 2018
%R 10.13718/j.cnki.xdzk.2018.08.011
%X $设<inline-formula>$ \mathscr{T}_X $</inline-formula>是非空集合<i>X</i>上的全变换半群，<i>E</i>是<i>X</i>上的（<i>n</i>，<i>m</i>）-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<inline-formula>$ \mathscr{T}_X $</inline-formula>的子半群.计算了变换半群<i>T</i><sub><i>E</i></sub>（<i>X</i>）的基数，并且在<i>n</i>=2，<i>m</i> ≥ 2和<i>n</i>=3，<i>m</i> ≥ 2的条件下，分别给出了<i>T</i><sub><i>E</i></sub>（<i>X</i>）的正则元个数的计算公式.$<br>$Let <inline-formula>$\mathscr{T}_X $</inline-formula> be a full transformation semigroup on the nonempty set <i>X</i>, and <i>E</i> be an (<i>n</i>, <i>m</i>)-type equivalence relation on <i>X</i>. 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 <i>T</i><sub><i>X</i></sub>. In this paper, we calculate the cardinality of the semigroup <i>T</i><sub><i>E</i></sub>(<i>X</i>) and present a formula for the number of its regular elements under the supposition that <i>n</i>=2, <i>m</i> ≥ 2 and <i>n</i>=3, <i>m</i> ≥ 2.
%K 变换半群
%K 等价关系
%K 正则元
%K 组合数&lt
%K br&gt
%K transformation semigroup
%K equivalence
%K regular element
%K combinatorial result
%U http://xbgjxt.swu.edu.cn/jsuns/html/jsuns/2018/8/20180811.htm