Relative ranks of some partial transformation semigroups

Authors : Ebru Yiğit, Gonca Ayik, Hayrullah Ayik

Pages : 2218-2225

View : 12 | Download : 6

Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :Let $P_{n}$, $T_{n}$, $I_{n}$, and $S_{n}$ be the partial transformation semigroup, the (full) transformation semigroup, the symmetric inverse semigroup, and the symmetric group on $X_{n}=\{1,\ldots ,n \}$, respectively. For $1\leq r\leq n-1$, let $PK_{n,r}$ be the subsemigroup consisting $\alpha\in P_{n}$ such that $|\im\alpha|\leq r$ and let $SPK_{n,r}=PK_{n,r}\setminus T_{n}$. In this paper, we first examine the subsemigroup $I_{n,r}=I_{n}\cup PK_{n,r}$ and we find the necessary and sufficient conditions for any nonempty subset of $PK_{n,r}$ to be a (minimal) relative generating set of the subsemigroup $I_{n,r}$ modulo $I_{n}$. Then we examine the subsemigroups $PI_{n,r}= SI_{n}\cup PK_{n,r}$ and $SI_{n,r}=SI_{n}\cup SPK_{n,r}$ for $1\leq r\leq n-1$ where $SI_{n}=I_{n}\setminus S_{n}$ and compute their relative rank.
Keywords : (Partial) transformation semigroup, symmetric inverse semigroup, symmetric group, (minimal) generating set, relative rank