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

Title & Authors
ON THE m-POTENT RANKS OF CERTAIN SEMIGROUPS OF ORIENTATION PRESERVING TRANSFORMATIONS
Zhao, Ping; You, Taijie; Hu, Huabi;

Abstract
It is known that the ranks of the semigroups $\small{\mathcal{SOP}_n}$, $\small{\mathcal{SPOP}_n}$ and $\small{\mathcal{SSPOP}_n}$ (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on $\small{X_n={1,2,{\ldots},n}}$, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of $\small{\mathcal{SOP}_n}$ and $\small{\mathcal{SSPOP}_n}$ 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 $\small{\mathcal{SOP}_n}$, $\small{\mathcal{SPOP}_n}$ and $\small{\mathcal{SSPOP}_n}$. Firstly, we characterize the structure of the minimal generating sets of $\small{\mathcal{SOP}_n}$. As applications, we obtain that the number of distinct minimal generating sets is $\small{(n-1)^nn!}$. Secondly, we show that, for $\small{1{\leq}m{\leq}n-1}$, the m-potent ranks of the semigroups $\small{\mathcal{SOP}_n}$ and $\small{\mathcal{SPOP}_n}$ are also n and 2n, respectively. Finally, we find that the 2-potent rank of $\small{\mathcal{SSPOP}_n}$ is n + 1.
Keywords
transformation;orientation-preserving;rank;idempotent rank;m-potent rank;
Language
English
Cited by
1.
On the (m, r)-potent ranks of certain semigroups of transformations, Journal of Algebra and Its Applications, 2016, 15, 01, 1650018
References
1.
G. Ayik, H. Ayik, Y. Unlu, and J. M. Howie, The structure of elements in finite full transformation semigroups, Bull. Aust. Math. Soc. 71 (2005), no. 1, 69-74.

2.
P. M. Catarino and P. M. Higgins, The monoid of orientation-preserving mappings on a chain, Semigroup Forum 58 (1999), no. 2, 190-206.

3.
V. H. Fernandes, G. M. S. Gomes, and M. M. Jesus, Congruences on monoids of transformations preserving the orientation on a finite chain, J. Algebra 321 (2009), no. 3, 743-757.

4.
G. M. S. Gomes and J. M. Howie, On the ranks of certain semigroups of order-preserving transformations, Semigroup Forum 45 (1992), no. 3, 272-282.

5.
J. M. Howie, Fundamentals of Semigroup Theory, Oxford, Oxford University Press, 1995.

6.
O. Sonmez, Combinatorics of elements in $T_n\;and\;PT_n$, Int. J. Algebra 4 (2010), no. 1-4, 45-51.

7.
O. Sonmez and Y. Unlu, m-potent elements in order-preserving transformation semigroups and ordered trees, Comm. Algebra 42 (2014), no. 1, 332-342.

8.
P. Zhao, On the ranks of certain semigroups of orientation preserving transformations, Comm. Algebra 39 (2011), no. 11, 4195-4205.

9.
P. Zhao, X. Bo, and Y. Mei, Locally maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups, Semigroup Forum 77 (2008), no. 2, 187-195.