Bulletin of the
Korean Mathematical Society BKMS

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