On the rank of transformation semigroup $T_{(n,m)}$

Authors : Kemal TOKER, Hayrullah AYIK

Pages : 1970-1977

View : 18 | Download : 9

Publication Date : 0000-00-00

Article Type : Research Paper

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 $$T_{insert ignore into journalissuearticles values(n,m);}=\{ \alpha \in T_{n} : X_{m} \alpha = X_{m}\} .$$ In this paper we research generating sets and the rank of $T_{insert ignore into journalissuearticles values(n,m);}$. In particular, we prove that $$rankinsert ignore into journalissuearticles values(T_{insert ignore into journalissuearticles values(n,m);});=\left\{ \begin{array}{lll} 2 & \mbox{ if }\, insert ignore into journalissuearticles values(n,m);=insert ignore into journalissuearticles values(2,1); \mbox{ or }insert ignore into journalissuearticles values(3,2);\\ 3 & \mbox{ if }\, insert ignore into journalissuearticles values(n,m);=insert ignore into journalissuearticles values(3,1); \mbox{ or } 4\leq n \mbox{ and } m=n-1\\ 4 & \mbox{ if }\, 4\leq n \mbox{ and } 1\leq m\leq n-2. \end{array}\right. $$ for $1\leq m\leq n-1$.
Keywords : Transformations, permutations, restricted image, generating set, rank