Research ArticlesNo Access

The $(p, q)$ -potent ranks of certain semigroups of transformations

Ping Zhao

School of Mathematical Sciences, Guizhou Normal University, Guiyang, GuiZhou Province 550001, P. R. China

E-mail Address: zhaoping731108@hotmail.com

Corresponding author.

Search for more papers by this author

Taijie You

School of Mathematical Sciences, Guizhou Normal University, Guiyang, GuiZhou Province 550001, P. R. China

E-mail Address: taijieyou@163.com

Search for more papers by this author

, and

Huabi Hu

Department of Medicine Bioengineering, Guiyang Medical University, GuiZhou Province 550004, P. R. China

E-mail Address: huhuabi@hotmail.com

Search for more papers by this author

https://doi.org/10.1142/S0219498817501389Cited by:0 (Source: Crossref)

Abstract

Let $𝒫_{n}$ and ${𝒮 𝒫}_{n}$ be the partial transformation and the strictly partial transformation semigroups on the finite set $X_{n} = {1, 2, \dots, n}$ . It is well known that the ranks of the semigroups $𝒫 (n, r) = {α \in 𝒫_{n} : | im (α) | \leq r}$ and $𝒮 𝒫 (n, r) = {α \in {𝒮 𝒫}_{n} : | im (α) | \leq r}$ are $S (n + 1, r + 1)$ , for $2 \leq r \leq n - 1$ , and $(r + 1) S (n, r + 1)$ , for $2 \leq r \leq n - 2$ , respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup $𝒫 (n, r)$ has the same value as the rank. Idempotent can be seen as a special case (with $p = q = 1$ ) of $(p, q)$ -potent. In this paper, we determine the $(p, q)$ -potent ranks, defined as the smallest number of $(p, q)$ -potents generating set, of the semigroups $𝒫 (n, r)$ , for $2 \leq r \leq n - 1$ , and $𝒮 𝒫 (n, r)$ , for $2 \leq r \leq n - 2$ .

Communicated by S. Sidki

Keywords:

AMSC: 20M20, 20M10

References

1. G. Ayık, H. Ayık, Y. Ünlü and J. M. Howie , The structure of elements in finite full transformation semigroups, Bull. Aust. Math. Soc. 71(1) (2005) 69–74. Crossref, Web of Science, Google Scholar
2. G. U. Garba , Idempotents in partial transformation semigroups, Proc. Roy. Soc. Edinburgh A 116 (1990) 359–366. Crossref, Web of Science, Google Scholar
3. G. U. Garba , On the nilpotent rank of partial transformation semigroups, Port. Math. 51(2) (1994) 163–172. Google Scholar
4. G. U. Garba , On the nilpotent ranks of certain semigroups of transformations, Glasg. Math. J. 36(1) (1994) 1–9. Crossref, Web of Science, Google Scholar
5. J. M. Howie , Fundamentals of Semigroup Theory (Oxford University Press, Oxford, 1995). Crossref, Google Scholar
6. O. Sönmez , Combinatorics of elements in $T_{n}$ and $P T_{n}$ , Int. J. Algebra 4(1–4) (2010) 45–51. Google Scholar