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Semigroups embeddable in hyperplane face monoids

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Abstract

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.

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Acknowledgements

The authors would like to thank Igor Dolinka for helpful correspondence on questions related to this paper and Mark Sapir for explaining in detail the results of his thesis [32] that impinge upon this paper.

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Stuart Margolis

  2. Département de Mathématiques—LaCIM, Université du Québec à Montréal, C.P. 8888, Succursale Centre-Ville, Montréal, Québec, H3C 3P8, Canada

    Franco Saliola

  3. Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, NY, 10031, USA

    Benjamin Steinberg

Authors
  1. Stuart Margolis
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  2. Franco Saliola
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  3. Benjamin Steinberg
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Correspondence to Stuart Margolis.

Additional information

Communicated by Nik Ruskuc.

Dedicated to the memory of John M. Howie.

The second author was supported in part by NSERC and FRQNT. The third author was supported in part by the Simon’s Foundation collaboration grant 245268. The first and third author’s research was supported by Grant No. 2012080 from the United States-Israel Binational Science Foundation (BSF).

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Margolis, S., Saliola, F. & Steinberg, B. Semigroups embeddable in hyperplane face monoids. Semigroup Forum 89, 236–248 (2014). https://doi.org/10.1007/s00233-013-9542-3

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