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Nilpotents in semigroups of partial one-to-one order-preserving mappings. (English) Zbl 0796.20058

Let \(A\) be a subset of an inverse semigroup \(S\). Denote by \(\langle A\rangle\) the inverse subsemigroup of \(S\) generated by \(A\). Let \(I_ n\) denote the symmetric inverse semigroup on a set of \(n\) elements and let \(IO_ n = \{\alpha \in I_ n: x \leq y\text{ implies }x\alpha \leq y\alpha\}\). Denote by \(N\) the set of nilpotent elements of \(IO_ n\). The depth of \(\langle N\rangle\) is the least integer \(k\) for which \(\langle N\rangle = N \cup N^ 2 \cup \cdots \cup N^ k\) and it is denoted by \(\Delta(\langle N\rangle)\). The elements in \(\langle N\rangle\) are characterized and it is shown that \(\Delta(\langle N\rangle) = 3\) whenever \(n \geq 3\). The rank of a semigroup \(S\) is the cardinality of a minimal set \(A\) for which \(\langle A\rangle = S\) and the nilpotent rank is the cardinality of a minimal set \(A\) of nilpotent elements for which \(\langle A\rangle = S\). It is shown that if \(r \leq n/2\), then the rank and the nilpotent rank of \(\{\alpha \in IO_ n: | \text{im }\alpha| \leq r\}\) are both equal to \({n\choose r} - 1\). If \(n/2 < r \leq n - 2\), then the rank and the nilpotent rank of \(\{\alpha \in IO_ n: | \text{im }\alpha| \leq r\) and \(\alpha \in \langle N\rangle\}\) are both equal to \({n\choose r} - {r-1\choose n-r} - 1\).

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
20M05 Free semigroups, generators and relations, word problems
11N45 Asymptotic results on counting functions for algebraic and topological structures
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References:

[1] Gomes, G. M. S. and J. M. Howie,Nilpotents in finite symmetric inverse semigroups, Proc. Edinburgh Math. Soc.30 (1986), 383–395. · Zbl 0629.20037 · doi:10.1017/S0013091500026778
[2] Gomes, G. M. S. and J. M. Howie,On the ranks of certain finite semigroups of transformations, Math. Proc. Cambridge Phil. Soc.101 (1987), 395–403. · Zbl 0622.20056 · doi:10.1017/S0305004100066780
[3] Howie, J. M., ”An introduction to semigroup theory,” Academic Press London, 1976. · Zbl 0355.20056
[4] Sullivan, R. P.,Semigroups generated by nilpotent transformations, J. Algebra110, (1987), 324–343. · Zbl 0626.20051 · doi:10.1016/0021-8693(87)90049-4
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