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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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