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The semigroups of order 9 and their automorphism groups

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Abstract

We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52 989 400 714 478 and up to isomorphism to be 105 978 177 936 292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148 195 347 518 186 distinct associative binary operations on an 8-element set and 38 447 365 355 811 944 462 on a 9-element set.

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Acknowledgements

We thank Robert Gray, James Mitchell and Steve Linton for helpful discussions, and James Mitchell and Csaba Schneider for comments on earlier versions of the paper.

The first author acknowledges the financial support from the doctoral program of the University of St Andrews and from the project PTDC/MAT/101993/2008 of Centro de Álgebra da Universidade de Lisboa, financed by FCT and FEDER.

The second author acknowledges the financial support by United Kingdom Engineering and Physical Sciences Research Council (EPSRC) grant EP/H004092/1.

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Authors and Affiliations

  1. Centro de Álgebra da Universidade de Lisboa, 1649-003, Lisboa, Portugal

    Andreas Distler

  2. School of Computer Science, University of St Andrews, KY16 9SX, St Andrews, UK

    Tom Kelsey

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  1. Andreas Distler
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  2. Tom Kelsey
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Correspondence to Andreas Distler.

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Communicated by Jean-Eric Pin.

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Distler, A., Kelsey, T. The semigroups of order 9 and their automorphism groups. Semigroup Forum 88, 93–112 (2014). https://doi.org/10.1007/s00233-013-9504-9

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