On the $m$-potent ranks of certain semigroups of orientation preserving transformations

摘要

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 $1leq mleq n-1$, 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 $n+1$.