Turkish Journal of Mathematics
DOI
10.3906/mat-1710-59
Abstract
Let $T_{n}$ and $S_{n}$ be the full transformation semigroup and the symmetric group on $X_{n}=\{1,\ldots,n\}$, respectively. For $n,m\in \mathbb{Z}^{+}$ with $m\le n-1$ let $$T_{(n,m)}=\{ \alpha \in T_{n} : X_{m} \alpha = X_{m}\} .$$ In this paper we research generating sets and the rank of $T_{(n,m)}$. In particular, we prove that $$rank(T_{(n,m)})=\left\{ \begin{array}{lll} 2 & \mbox{ if }\, (n,m)=(2,1) \mbox{ or }(3,2)\\ 3 & \mbox{ if }\, (n,m)=(3,1) \mbox{ or } 4\leq n \mbox{ and } m=n-1\\ 4 & \mbox{ if }\, 4\leq n \mbox{ and } 1\leq m\leq n-2. \end{array}\right. $$ for $1\leq m\leq n-1$.
Keywords
Transformations, permutations, restricted image, generating set, rank
First Page
1970
Last Page
1977
Recommended Citation
TOKER, KEMAL and AYIK, HAYRULLAH
(2018)
"On the rank of transformation semigroup $T_{(n,m)}$,"
Turkish Journal of Mathematics: Vol. 42:
No.
4, Article 33.
https://doi.org/10.3906/mat-1710-59
Available at:
https://journals.tubitak.gov.tr/math/vol42/iss4/33
