%0 Journal Article
%T 半群<i>S<sub>n</sub></i>(<i>k</i>)的秩<br>Ranks of the Semigroup <i>S<sub>n</sub></i>(<i>k</i>)
%A 张传军
%A 朱华伟&lt
%A br&gt
%A ZHANG Chuan-jun
%A ZHU Hua-wei
%J 西南大学学报(自然科学版)
%D 2017
%R 10.13718/j.cnki.xdzk.2017.06.010
%X 设Sin<i>g</i><sub>n</sub>是[<i>n</i>]上的奇异变换半群.对任意1≤<i>k</i>≤<i>n</i>-1, 研究半群
<inline-formula>${S_n}\left( k \right) = \left\{ {\alpha  \in {\text{Sin}}{{\text{g}}_n}:\forall x \in \left[ n \right],x \leqslant k \Rightarrow x\alpha  \leqslant k} \right\}$</inline-formula>



证明了<i>S<sub>n</sub></i>(<i>k</i>)是由秩为<i>n</i>-1的幂等元生成的, 并得到半群<i>S<sub>n</sub></i>(<i>k</i>)(<i>k</i>≠2) 的秩和幂等元秩都为<inline-formula>$\frac{{n(n - 1)}}{2}$</inline-formula>.同时, 得到了半群<i>S<sub>n</sub></i>(2) 的秩和幂等元秩都为<inline-formula>$\frac{{n(n - 1)}}{2}$</inline-formula>.<br>Let Sin<i>g<sub>n</sub></i> be a singular transformation semigroup on [<i>n</i>]. For an arbitrary integer 1≤<i>k</i>≤<i>n</i>-1, the rank and idempotent rank of the semigroup <i>S<sub>n</sub></i>(<i>k</i>)={<i>α</i>∈Sin<i>g<sub>n</sub></i>:？<i>x</i>∈[<i>n</i>], <i>x</i>≤<i>k</i><inline-formula>$ \Rightarrow  $</inline-formula><i>xα</i>≤<i>k</i>} are studied. We show that the semigroup <i>S<sub>n</sub></i>(<i>k</i>) is generated by the idempotents of rank <i>n</i>-1, and obtain that the rank and idempotent rank of the semigroup <i>S<sub>n</sub></i>(<i>k</i>)(<i>k</i>≠2) are both equal to <inline-formula>$ \frac{n(n-1)}{2} $</inline-formula>, and the rank and idempotent rank of the semigroup <i>S<sub>n</sub></i>(2) are both equal to <inline-formula>$ \frac{n(n-1)}{2} $</inline-formula>
%K 奇异变换半群
%K 幂等元秩
%K 秩&lt
%K br&gt
%K singular transformation semigroup
%K idempotent rank
%K rank
%U http://xbgjxt.swu.edu.cn/jsuns/html/jsuns/2017/6/201706010.htm