PORTUGALIAE MATHEMATICA Vol. 53, No. 1, pp. 2334 (1996) 

On the Ranks of Certain Finite Semigroups of OrderDecreasing TransformationsAbdullahi UmarDepartment of Mathematics, University of Abuja,Federal Capital Territory  NIGERIA Abstract: Let $T_{n}$ be the full transformation semigroup on a totally ordered finite set with $n$ elements and let $K^{}(n,r)=\{\alpha\in T_{n}: x\,\alpha\le x$ and $\IM\alpha\le r\}$, be the subsemigroup of $T_{n}$ consisting of all decreasing maps $\alpha$, for which $\IM\alpha\le r$. Similarly, let $I_{n}$ be the partial oneone transformation semigroup on a totally ordered finite set with $n$ elements and let $L^{}(n,r)=\{\alpha\in I_{n}: x\,\alpha\le x$ and $\IM\alpha\le r\}\cup\{\emptyset\}$, be the subsemigroup of $I_{n}$ consisting of all decreasing partial oneone maps $\alpha$ (including the empty or zero map), for which $\!\IM\alpha\le r$. If we define the rank of a finite semigroup $S$ as the cardinal of a minimal generating set of $S$, then in this paper it is shown that the Rees quotient semigroups $P_{r}^{}=K^{}(n,r)/K^{}(n,r1)$ (for $n\ge3$ and $r\ge2$) and $Q_{r}^{}=L^{}(n,r)/L^{}(n,r1)$ (for $n\ge2$ and $r\ge1$) each admits a unique minimal generating set. Further, it is shown that for $1\le r\le n1$, $\rank P_{r}^{}=S(n,r)$, the Stirling number of the second kind, and for $1\le r\le n1$ $$ \rank Q_{r}^{}=\biggl({n\atop r1}\biggr)\, \frac{\Bigl[(nr)\,(r+1)+1\Bigr]}{1}. $$ Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1996 Sociedade Portuguesa de Matemática
