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Enumeration of finite inverse semigroups

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Abstract

We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order \(n\le 15\). This improves considerably on the previous highest-known value S(9). We also give a related algorithm for the enumeration up to isomorphism of the finite inverse semigroups S with a given underlying semilattice of idempotents E, a given restriction of Green’s \(\mathrel {\mathcal {D}}\)-relation on S to E, and a given list of maximal subgroups of S associated to the elements of E.

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Notes

  1. Cayley tables and Sage code available at http://www.shsu.edu/mem037/ISGs.html.

  2. Lattices available at http://www.shsu.edu/mem037/Lattices.html.

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Acknowledgements

We are grateful to Sam Houston State University (SHSU) and the IT@Sam department at SHSU for their assistance in building and maintaining the server on which we obtained our computational results. We are also thankful to the anonymous referee whose suggestions have helped improve the presentation and readability of the paper.

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  1. Department of Mathematics and Statistics, Sam Houston State University, Box 2206, Huntsville, TX, 77341-2206, USA

    Martin E. Malandro

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  1. Martin E. Malandro
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Correspondence to Martin E. Malandro.

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Communicated by Benjamin Sternberg.

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Malandro, M.E. Enumeration of finite inverse semigroups. Semigroup Forum 99, 679–723 (2019). https://doi.org/10.1007/s00233-019-10054-9

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