## On the ranks of certain finite semigroups of order-decreasing transformations.(English)Zbl 0844.20050

Let $$X$$ be a totally ordered finite set with $$n$$ elements and let $$K^- (n,r)$$ be the semigroup, under composition, of all selfmaps $$\alpha$$ of $$X$$ such that $$x\alpha \leq x$$ and $$|\text{Im } \alpha |\leq r$$. Let $$L^-(n,r)$$ be the semigroup consisting of the empty map together with all partial one-to-one maps $$\alpha$$ on $$X$$ such that $$x\alpha \leq x$$ and $$|\text{Im } \alpha|\leq r$$. Let $$P^-_r = K^-(n,r)/K^-(n,r-1)$$ for $$n \geq 3$$ and $$r \geq 2$$ and let $$Q_r^- = L^-(n,r)/L^-(n,r - 1)$$ for $$n \geq 2$$ and $$r \geq 1$$. The author shows that both $$P^-_r$$ and $$Q^-_r$$ have a unique generating set. He further shows that $$\text{rank } P^-_r = S(n,r)$$ the Stirling number of the second kind where the rank, $$\text{rank }S$$, of a semigroup $$S$$ is defined to be the least of the cardinal numbers of all the generating sets of $$S$$. Finally, he determines the rank of $$Q^-_r$$ as well.

### MSC:

 20M20 Semigroups of transformations, relations, partitions, etc. 20M05 Free semigroups, generators and relations, word problems
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