On the Ranks of Certain Finite Semigroups of Order-Decreasing Transformations

Abdullahi Umar

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 one-one 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 one-one 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 $P_{r}^{-}=K^{-}(n,r)/K^{-}(n,r-1)$ (for $n\ge3$ and $r\ge2$) and $Q_{r}^{-}=L^{-}(n,r)/L^{-}(n,r-1)$ (for $n\ge2$ and $r\ge1$) each admits a unique minimal generating set. Further, it is shown that for $1\le r\le n-1$, $\rank P_{r}^{-}=S(n,r)$, the Stirling number of the second kind, and for $1\le r\le n-1$ $$ \rank Q_{r}^{-}=\biggl({n\atop r-1}\biggr)\, \frac{\Bigl[(n-r)\,(r+1)+1\Bigr]}{1}. $$

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