On the $m$-potent ranks of certain semigroups of orientation preserving transformations
Bull. Korean Math. Soc. 2014 Vol. 51, No. 6, 1841-1850
Printed November 30, 2014
Ping Zhao, Taijie You, and Huabi Hu
Guiyang Medical College, Guizhou Normal University, Guiyang Medical College
Abstract : It is known that the ranks of the semigroups $\SOP_n$, $\SPOP_n$ and $\SSPOP_n$ (the semigroups of orientation preserving singular selfmaps, partial and strictly partial transformations on $X_n=\{1,2,\dots,n\}$, respectively) are $n$, $2n$ and $n+1$, respectively. The \emph{idempotent rank}, defined as the smallest number of idempotent generating set, of $\SOP_n$ and $\SSPOP_n$ are the same value as the rank, respectively. Idempotent can be seen as a special case (with $m=1$) of $m$-potent. In this paper, we investigate the $m$-potent ranks, defined as the smallest number of $m$-potent generating set, of the semigroups $\SOP_n$, $\SPOP_n$ and $\SSPOP_n$. Firstly, we characterize the structure of the minimal generating sets of $\SOP_n$. As applications, we obtain that the number of distinct minimal generating sets is $(n-1)^nn!$. Secondly, we show that, for $1\leq m\leq n-1$, the $m$-potent ranks of the semigroups $\SOP_n$ and $\SPOP_n$ are also $n$ and $2n$, respectively. Finally, we find that the $2$-potent rank of $\SSPOP_n$ is $n+1$.
Keywords : transformation, orientation-preserving, rank, idempotent rank, $m$-potent rank
MSC numbers : 20M20, 20M10
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