Abstract
J.M. Howie proved that \(\operatorname {Sing}_{n}\), the semigroup of all singular mappings of {1,…,n} into itself, is generated by its idempotents of defect 1 (in J. London Math. Soc. 41, 707–716, 1966). He also proved that if n≥3 then a minimal generating set for \(\operatorname {Sing}_{n}\) contains n(n−1)/2 transformations of defect 1 (in Gomes and Howie, Math. Proc. Camb. Philos. Soc. 101. 395–403, 1987). In this paper we find necessary and sufficient conditions for any set for transformations of defect 1 in \(\operatorname {Sing}_{n}\) to be a (minimal) generating set for \(\operatorname {Sing}_{n}\).
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The authors would like to thank Nesin Mathematics Village (Şirince-Izmir, Turkey) and its supporters for providing a peaceful environment where some parts of this research was carried out.
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Communicated by Steve Pride.
Dedicated to the memory of John Mackintosh Howie (1936–2011).
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Ayık, G., Ayık, H., Bugay, L. et al. Generating sets of finite singular transformation semigroups. Semigroup Forum 86, 59–66 (2013). https://doi.org/10.1007/s00233-012-9379-1
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DOI: https://doi.org/10.1007/s00233-012-9379-1