Turkish Journal of Mathematics
Abstract
Let $I_{n}$ and $S_{n}$ be the symmetric inverse semigroup and the symmetric group on a finite chain $X_{n}=\{1,\ldots ,n \}$, respectively. Also, let $I_{n,r}= \{ \alpha \in I_{n}: im(\alpha) \leq r\}$ for $1\leq r\leq n-1$. For any $\alpha\in I_n$, if $\alpha\neq \alpha^2=\alpha^4$ then $\alpha$ is called a quasi-idempotent. In this paper, we show that the quasi-idempotent rank of $I_{n,r}$ (both as a semigroup and as an inverse semigroup) is $\binom{n}{2}$ if $r=2$, and $\binom{n}{r}+1$ if $r\geq 3$. The quasi-idempotent rank of $I_{n,1}$ is $n$ (as a semigroup) and $n-1$ (as an inverse semigroup).
DOI
10.3906/mat-2001-3
Keywords
Symmetric inverse semigroup, symmetric group, quasi-idempotent, rank
First Page
281
Last Page
287
Recommended Citation
BUGAY, L (2021). Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups. Turkish Journal of Mathematics 45 (1): 281-287. https://doi.org/10.3906/mat-2001-3
