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ON THE m-POTENT RANKS OF CERTAIN SEMIGROUPS OF ORIENTATION PRESERVING TRANSFORMATIONS

  • Zhao, Ping (School of Mathematics and Computer Science Guizhou Normal University, Mathematics Teaching & Research Section Guiyang Medical College) ;
  • You, Taijie (School of Mathematics and Computer Science Guizhou Normal University) ;
  • Hu, Huabi (Department of Medicine Bioengineering Guiyang Medical College)
  • Received : 2013.11.28
  • Published : 2014.11.30

Abstract

It is known that the ranks of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$ (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on $X_n={1,2,{\ldots},n}$, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of $\mathcal{SOP}_n$ and $\mathcal{SSPOP}_n$ 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 $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$. Firstly, we characterize the structure of the minimal generating sets of $\mathcal{SOP}_n$. 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 $\mathcal{SOP}_n$ and $\mathcal{SPOP}_n$ are also n and 2n, respectively. Finally, we find that the 2-potent rank of $\mathcal{SSPOP}_n$ is n + 1.

Keywords

Acknowledgement

Supported by : Natural Science Foundation of Guizhou, National Natural Science Foundation of China

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Cited by

  1. On the (m, r)-potent ranks of certain semigroups of transformations vol.15, pp.01, 2016, https://doi.org/10.1142/S0219498816500183